in fig1 to 3 and fig7 to 9 , there are depicted the lens forms and the lens movements represented by the numerical data for numerical examples 1 to 6 of the invention . in the drawings , reference numeral 1 denotes a first lens unit arranged on the object side ( long conjugate side ), having a negative refractive power and , here , comprising a negative meniscus lens convex toward the object side . reference numeral 2 denotes a second lens unit likewise comprising a negative meniscus lens convex toward the object side . reference numeral 3 denotes a third lens unit having a positive refractive power and comprising either a cemented lens composed of a positive lens having a strong convex surface facing the image side ( short conjugate side ) and a negative lens , or a bi - convex single lens . reference numeral 4 denotes a fourth lens unit having a negative refractive power and comprising a positive lens having a strong convex surface facing the image side followed , after a spacing therefrom , by a negative lens having a strong concave surface facing the object side . the first lens unit 1 to the fourth lens unit 4 move toward the object side when zooming from the wide - angle end to the telephoto end . in the zoom lenses of fig1 to 3 and fig9 as zooming is effected from the wide - angle end w to the telephoto end t , the separation between the first and second lens units increases , the separation between the second and third lens units decreases , and the separation between the third and fourth lens units decreases . in the zoom lenses of fig7 and 8 , the separation between the first and second lens units decreases , the separation between the second and third lens units decreases , and the separation between the third and fourth lens units decreases . incidentally , the arrows in the drawings schematically show the direction of movement and do not trace the loci of motion . further , of the embodiments , there are ones in which the first and third lens units move in unison during zooming . in addition , the zoom lens of each embodiment of the invention is made to satisfy the following conditions : where d iw is the separation between the i - th and ( i + 1 ) st lens units in the wide - angle end of the zooming range , and d it is the separation between the i - th and ( i + 1 ) st lens units in the telephoto end . the inequalities ( 1 ) and ( 2 ) are to regulate the varied separation between the third and fourth lens units at the wide - angle end and the telephoto end in relation to the varied separations between the second and third lens units and between the first and second lens units , respectively . when each condition is violated , not only the various aberrations become difficult to correct , but also the size of the entire lens system increases objectionably . further , it is desirable that the zoom lenses of fig1 to 3 satisfy at least one of the following conditions : where fi is the focal length of the i - th lens unit and f w is the shortest focal length of the entire lens system . the meanings of the limit values of the inequalities ( 3 ) to ( 6 ) are explained below . the inequality ( 3 ) and the inequality ( 4 ) are concerned respectively with the ratios of the focal lengths of the first and second lens units to the focal length of the third lens unit . when the refractive power of the first or second lens unit is too weak as exceeding the lower limit of each condition , the longitudinal chromatic aberration becomes difficult to correct . when the refractive power of the third lens unit is too weak as exceeding the upper limit , it becomes difficult to secure the back focal distance in the wide - angle end . the inequality ( 5 ) is concerned with the ratio of the focal length of the third lens unit to the shortest focal length of the entire lens system . when the refractive power of the third lens unit is too strong as exceeding the lower limit , this lens unit becomes highly sensitive , thus becoming difficult to manufacture . so , it is no good . when the refractive power of the third lens unit is too weak as exceeding the upper limit of the inequalities of condition ( 5 ), it becomes difficult to secure the back focal distance in the wide - angle end . the inequality ( 6 ) is concerned with the ratio of the focal length of the fourth lens unit to the shortest focal length of the entire lens system . when the refractive power of the fourth lens unit is weaker than the lower limit , a predetermined zoom ratio becomes difficult to obtain , or it interferes with the third lens unit in the telephoto end . so , it is no good . when the refractive power of the fourth lens unit is stronger than the upper limit , it becomes difficult to correct for a variation of the various aberrations . the numerical examples are described below . the character r represents the radius of curvature , d the lens thickness or lens separation , n the refractive index for the d - line and ν the abbe number . the aspheric surface is expressed by the following equation for the axial deviation x in terms of the radius r of the osculating sphere and the height h from the optical axis : ______________________________________numerical example 1 ( fig1 ( a ), 4 ( b ) and 4 ( c )) ______________________________________f = 36 . 19 - 101 . 5 fno = 1 : 4 . 1 - 8 . 8 2ω = 61 . 8 °- 24 . 0 ° r 1 = 50 . 000 d 1 = 1 . 50 n 1 = 1 . 603111 ν 1 = 60 . 7r 2 = 45 . 651 d 2 = variabler 3 = 24 . 178 d 3 = 1 . 50 n 2 = 1 . 77250 ν 2 = 49 . 6r 4 = 19 . 520 d 4 = variabler 5 = 95 . 414 d 5 = 5 . 00 n 3 = 1 . 49700 ν 3 = 81 . 6r 6 = 9 . 695 d 6 = 1 . 50 n 4 = 1 . 83400 ν 4 = 37 . 2r 7 = - 12 . 299 d 7 = 1 . 00r 8 = stop d 8 = variabler 9 = - 68 . 986 d 9 = 3 . 50 n 5 = 1 . 59551 ν 5 = 39 . 2r 10 = - 26 . 500 d 10 = 4 . 64r 11 = - 13 . 971 d 11 = 1 . 70 n 6 = 1 . 69680 ν 6 = 55 . 5r 12 = 206 . 234______________________________________variable focal lengthseparation 36 . 20 63 . 84 101 . 50______________________________________d 2 2 . 00 1 . 18 1 . 93d 4 8 . 32 8 . 14 7 . 38d 8 16 . 65 7 . 09 2 . 51______________________________________ r 3 is aspherical . a = 0 b = - 4 . 43765 × 10 . sup .- 5 c = - 2 . 81409 × 10 . sup .- 7 d = - 1 . 36697 × 10 . sup .- 9 e = - 9 . 28334 × 10 . sup .- 12 r 7 is aspherical . a = 0 b = 1 . 40440 × 10 . sup .- 5 c = 5 . 14255 × 10 . sup .- 8 d = 0 , e = 0 r 10 is aspherical . a = 0 b = - 3 . 21458 × 10 . sup .- 5 c = - 1 . 00841 × 10 . sup .- 7 d = - 2 . 99727 × 10 . sup .- 10 e = - 3 . 58134 × 10 . sup .- 12______________________________________ ______________________________________numerical example 2 ( fig2 ( a ), 5 ( b ) and 5 ( c )) ______________________________________f = 36 . 2 - 101 . 5 fno = 1 : 4 . 1 - 8 . 8 2ω = 61 . 8 °- 24 . 0 ° r 1 = 50 . 000 d 1 = 1 . 50 n 1 = 1 . 71300 ν 1 = 53 . 8r 2 = 46 . 140 d 2 = variabler 3 = 22 . 158 d 3 = 1 . 50 n 2 = 1 . 85026 ν 2 = 32 . 3r 4 = 18 . 121 d 4 = variabler 5 = 102 . 458 d 5 = 5 . 00 n 3 = 1 . 51633 ν 3 = 64 . 2r 6 = - 9 . 096 d 6 = 1 . 50 n 4 = 1 . 83400 ν 4 = 37 . 2r 7 = - 12 . 188 d 7 = 1 . 00r 8 = stop d 8 = variabler 9 = - 57 . 224 d 9 = 3 . 50 n 5 = 1 . 56732 ν 5 = 42 . 8r 10 = - 25 . 174 d 10 = 4 . 86r 11 = - 13 . 760 d 11 = 1 . 70 n 6 = 1 . 69680 ν 6 = 55 . 5r 12 = 620 . 901______________________________________variable focal lengthseparation 36 . 20 63 . 84 101 . 51______________________________________d 2 1 . 00 1 . 11 1 . 78d 4 8 . 53 8 . 42 7 . 75d 8 16 . 84 7 . 24 2 . 64______________________________________ r 3 is aspherical . a = 0 b = - 3 . 94324 × 10 . sup .- 5 c = - 2 . 31323 × 10 . sup .- 7 d = - 1 . 42198 × 10 . sup .- 9 e = - 8 . 95882 × 10 . sup .- 12 r 7 is aspherical . a = 0 b = 1 . 08432 × 10 . sup .- 5 c = 3 . 94859 × 10 . sup .- 9 d = 0 , e = 0 r 10 is aspherical . a = 0 b = - 3 . 23425 × 10 . sup .- 5 c = 2 . 25379 × 10 . sup .- 8 d = - 1 . 4729 × 10 . sup .- 9 e = 3 . 38991 × 10 . sup .- 12______________________________________ ______________________________________numerical example 3 ( fig3 ( a ), 6 ( b ) and 6 ( c )) ______________________________________f = 36 . 19 - 101 . 52 fno = 1 : 4 . 1 - 8 . 75 2ω = 61 . 8 °- 24 . 0 ° r 1 = 50 . 000 d 1 = 1 . 50 n 1 = 1 . 71300 ν 1 = 53 . 8r 2 = 46 . 140 d 2 = variabler 3 = 24 . 306 d 3 = 1 . 50 n 2 = 1 . 83400 ν 2 = 37 . 2r 4 = 19 . 736 d 4 = variabler 5 = 110 . 499 d 5 = 5 . 00 n 3 = 1 . 51633 ν 3 = 64 . 2r 6 = - 9 . 591 d 6 = 1 . 50 n 4 = 1 . 85026 ν 4 = 32 . 3r 7 = - 12 . 403 d 7 = 1 . 00r 8 = stop d 8 = variabler 9 = - 66 . 429 d 9 = 3 . 50 n 5 = 1 . 56732 ν 5 = 42 . 8r 10 = - 25 . 943 d 10 = 4 . 69r 11 = - 13 . 927 d 11 = 1 . 70 n 6 = 1 . 69680 ν 6 = 55 . 5r 12 = 258 . 899______________________________________variable focal lengthseparation 36 . 20 63 . 84 101 . 52______________________________________d 2 1 . 00 1 . 17 1 . 85d 4 8 . 26 8 . 09 7 . 41d 8 16 . 62 7 . 10 2 . 53______________________________________ r 3 is aspherical . a = 0 b = - 4 . 19556 × 10 . sup .- 5 c = - 2 . 69123 × 10 . sup .- 7 d = - 1 . 14486 × 10 . sup .- 9 e = - 1 . 06748 × 10 . sup .- 11 r 7 is aspherical . a = 0 b = 1 . 21244 × 10 . sup .- 5 c = 3 . 43521 × 10 . sup .- 8 d = 0 , e = 0 r 10 is aspherical . a = 0 b = - 3 . 33544 × 10 . sup .- 5 c = - 5 . 37753 × 10 . sup .- 8 d = - 4 . 17385 × 10 . sup .- 10 e = - 6 . 26109 × 10 . sup .- 13______________________________________ table 1______________________________________ example 1 example 2 example 3______________________________________d . sub . 3w - d . sub . 3t 14 . 14 14 . 20 14 . 093 ( d . sub . 2w - d . sub . 2t ) 2 . 80 2 . 34 2 . 553 ( d . sub . 1t - d . sub . 1w ) 2 . 80 2 . 34 2 . 55f . sub . 1 / f . sub . 3 - 39 . 29 - 39 . 63 - 39 . 54f . sub . 2 / f . sub . 3 - 5 . 99 - 5 . 59 - 5 . 85f . sub . 3 / f . sub . w 0 . 70 0 . 70 0 . 70f . sub . 4 / f . sub . w - 0 . 76 - 0 . 76 - 0 . 75______________________________________ there are next shown numerical examples in which the third lens unit is constructed with a positive single lens having a strong convex surface facing the image side . ______________________________________numerical example 4 ( fig7 ( a ), 10 ( b ) and 10 ( c )) ______________________________________f = 36 . 2 - 101 . 5 fno = 1 : 4 . 6 - 8 . 8 2ω = 61 . 8 °- 24 . 0 ° r 1 = 19 . 418 d 1 = 1 . 50 n 1 = 1 . 77250 ν 1 = 49 . 6r 2 = 16 . 180 d 2 = variabler 3 = 30 . 730 d 3 = 1 . 50 n 2 = 1 . 83400 ν 2 = 37 . 2r 4 = 18 . 698 d 4 = variabler 5 = 23 . 309 d 5 = 5 . 00 n 3 = 1 . 49700 ν 3 = 81 . 6r 6 = - 14 . 879 d 6 = 1 . 00r 7 = stop d 7 = variabler 8 = - 57 . 616 d 8 = 3 . 20 n 4 = 1 . 60342 ν 4 = 38 . 0r 9 = - 24 . 757 d 9 = 4 . 68r 10 = - 12 . 817 d 10 = 1 . 70 n 5 = 1 . 69680 ν 5 = 55 . 5r 11 = - 455 . 372______________________________________variable focal lengthseparation 36 . 20 64 . 34 101 . 51______________________________________d 2 8 . 53 8 . 66 8 . 49d 4 1 . 72 1 . 59 1 . 75d 7 16 . 77 6 . 93 2 . 11______________________________________ r 1 is aspherical . a = 0 b = - 3 . 48999 × 10 . sup .- 5 c = - 2 . 31998 × 10 . sup .- 7 d = - 9 . 41482 × 10 . sup .- 10 e = - 5 . 02295 × 10 . sup .- 12______________________________________ ______________________________________numerical example 5 ( fig8 ( a ), 11 ( b ) and 11 ( c )) ______________________________________f = 36 . 2 - 101 . 5 fno = 1 : 4 . 6 - 8 . 8 2ω = 61 . 8 °- 24 . 0 ° r 1 = 18 . 582 d 1 = 1 . 50 n 1 = 1 . 77250 ν 1 = 49 . 6r 2 = 15 . 811 d 2 = variabler 3 = 33 . 582 d 3 = 1 . 50 n 2 = 1 . 80610 ν 2 = 41 . 0r 4 = 18 . 105 d 4 = variabler 5 = 21 . 679 d 5 = 5 . 00 n 3 = 1 . 49700 ν 3 = 81 . 6r 6 = - 14 . 535 d 6 = 1 . 00r 7 = stop d 7 = variabler 8 = - 63 . 081 d 8 = 3 . 20 n 4 = 1 . 58144 ν 4 = 40 . 8r 9 = - 25 . 420 d 9 = 4 . 68r 10 = - 12 . 950 d 10 = 1 . 70 n 5 = 1 . 69680 ν 5 = 55 . 5r 11 = - 985 . 684______________________________________variable focal lengthseparation 36 . 20 64 . 25 101 . 50______________________________________d 2 10 . 23 10 . 10 9 . 92d 4 1 . 42 1 . 56 1 . 73d 7 16 . 66 6 . 79 2 . 06______________________________________ r 1 is aspherical . a = 0 b = - 2 . 54915 × 10 . sup .- 5 c = - 2 . 37724 × 10 . sup .- 7 d = 3 . 0728 × 10 . sup .- 10 e = - 1 . 06680 × 10 . sup .- 11 r 3 is aspherical . a = 0 b = - 1 . 65774 × 10 . sup .- 5 c = - 1 . 38823 × 10 . sup .- 7 d = 0 , e = 0 r 9 is aspherical . a = 0 b = - 2 . 58518 × 10 . sup .- 5 c = - 1 . 72514 × 10 . sup .- 7 d = 6 . 63195 × 10 . sup .- 10 e = - 7 . 02040 × 10 . sup .- 12______________________________________ ______________________________________numerical example 6 ( fig9 ( a ), 12 ( b ) and 12 ( c )) ______________________________________f = 36 . 2 - 101 . 5 fno = 1 : 4 . 1 - 8 . 8 2ω = 61 . 8 °- 24 . 0 ° r 1 = 50 . 000 d 1 = 1 . 50 n 1 = 1 . 80518 ν 1 = 25 . 4r 2 = 46 . 447 d 2 = variabler 3 = 28 . 468 d 3 = 1 . 50 n 2 = 1 . 85026 ν 2 = 32 . 3r 4 = 21 . 758 d 4 = variabler 5 = 60 . 136 d 5 = 6 . 00 n 3 = 1 . 43387 ν 3 = 95 . 1r 6 = - 13 . 197 d 6 = 1 . 00r 7 = stop d 7 = variabler 8 = - 166 . 800 d 8 = 3 . 50 n 4 = 1 . 54814 ν 4 = 45 . 8r 9 = - 31 . 753 d 9 = 4 . 83r 10 = - 14 . 455 d 10 = 1 . 70 n 5 = 1 . 69680 ν 5 = 55 . 5r 11 = 147 . 439______________________________________variable focal lengthseparation 36 . 20 64 . 26 101 . 55______________________________________d 2 1 . 00 0 . 93 1 . 56d 4 8 . 86 8 . 93 8 . 30d 7 17 . 11 6 . 86 2 . 12______________________________________ r 3 is aspherical . a = 0 b = - 4 . 15008 × 10 . sup .- 5 c = - 2 . 15803 × 10 . sup .- 7 d = - 1 . 37123 × 10 . sup .- 9 e = - 5 . 32786 × 10 . sup .- 12 r 6 is aspherical . a = 0 b = 3 . 22540 × 10 . sup .- 5 c = 1 . 10695 × 10 . sup .- 7 d = 0 , e = 0 r 9 is aspherical . a = 0 b = - 3 . 64976 × 10 . sup .- 5 c = - 5 . 06977 × 10 . sup .- 8 d = - 5 . 04985 × 10 . sup .- 10 e = - 1 . 03634 × 10 . sup .- 12______________________________________ table 2______________________________________ example 4 example 5 example 6______________________________________ | d . sub . 3w - d . sub . 3t | 14 . 6 14 . 59 14 . 993 | d . sub . 2w - d . sub . 2t | 0 . 33 0 . 95 1 . 693 | d . sub . 1t - d . sub . 1w | 0 . 33 0 . 95 1 . 69______________________________________ fig1 and fig1 are embodiments different from the above - described ones in that the signs of the refractive powers of the second and third lens units are inverted . the first to the third lens units each comprise a single lens . the first lens unit 1 is a negative meniscus lens convex toward the object side . the second lens unit 2 is a bi - convex lens having a strong convex surface facing the image side . the third lens unit is a negative meniscus lens convex toward the image side . the fourth lens unit 4 is composed of a positive meniscus lens convex toward the image side and a negative lens having a strong concave surface facing the object side . for the purpose of zooming from the wide - angle end to the telephoto end , the first to fourth lens units move toward the object side in such a way that the separation between the first lens unit 1 and the second lens unit 2 decreases , the separation between the second lens unit 2 and the third lens unit 3 increases and the separation between the third lens unit 3 and the fourth lens unit 4 decreases . for these embodiments 7 and 8 , the above - described conditions ( 1 ) and ( 2 ) also apply . ______________________________________numerical example 7 ( fig1 , 15 ( a ), 15 ( b ) and 15 ( c )) ______________________________________f = 36 . 15 - 101 . 98 fno = 1 : 4 . 6 - 8 . 8 2ω = 61 . 8 °- 24 . 0 ° r 1 = 24 . 178 d 1 = 1 . 50 n 1 = 1 . 69680 ν 1 = 55 . 5r 2 = 16 . 755 d 2 = variabler 3 = stop d 3 = 1 . 00r 4 = 48 . 333 d 4 = 5 . 00 n 2 = 1 . 43387 ν 2 = 95 . 1r 5 = - 12 . 720 d 5 = variabler 6 = - 26 . 023 d 6 = 1 . 50 n 3 = 1 . 85026 ν 3 = 32 . 3r 7 = - 28 . 370 d 7 = variabler 8 = - 114 . 508 d 8 = 4 . 00 n 4 = 1 . 54814 ν 4 = 45 . 8r 9 = - 30 . 116 d 9 = 4 . 64r 10 = - 17 . 252 d 10 = 1 . 70 n 5 = 1 . 77250 ν 5 = 49 . 6r 11 = 142 . 249______________________________________variable focal lengthseparation 36 . 16 64 . 09 101 . 98______________________________________d 2 10 . 70 10 . 41 9 . 69d 5 1 . 50 1 . 78 2 . 50d 7 17 . 52 6 . 72 0 . 95______________________________________ r 1 is aspherical . a = 0 b = - 4 . 1428 × 10 . sup .- 5 c = - 2 . 07199 × 10 . sup .- 7 d = - 1 . 80897 × 10 . sup .- 9 r 5 is aspherical . a = 0 b = 5 . 11587 × 10 . sup .- 5 c = 1 . 71220 × 10 . sup .- 7 d = 0 , e = 0 r 9 is aspherical . a = 0 b = - 2 . 85035 × 10 . sup .- 5 c = 1 . 3479 × 10 . sup .- 8 d = - 1 . 88032 × 10 . sup .- 10 e = - 3 . 14063 × 10 . sup .- 13______________________________________ ______________________________________numerical example 8 ( fig1 , 16 ( a ), 16 ( b ) and 16 ( c )) ______________________________________f = 36 . 2 - 102 . 0 fno = 1 : 4 . 6 - 8 . 8 2ω = 61 . 8 °- 24 . 0 ° r 1 = 24 . 178 d 1 = 1 . 50 n 1 = 1 . 77250 ν 1 = 49 . 6r 2 = 17 . 280 d 2 = variabler 3 = stop d 3 = 1 . 00r 4 = 62 . 072 d 4 = 5 . 00 n 2 = 1 . 49700 ν 2 = 81 . 6r 5 = - 14 . 223 d 5 = variabler 6 = - 26 . 049 d 6 = 1 . 50 n 3 = 1 . 80518 ν 3 = 25 . 4r 7 = - 28 . 375 d 7 = variabler 8 = - 112 . 885 d 8 = 4 . 00 n 4 = 1 . 54814 ν 4 = 45 . 8r 9 = - 29 . 063 d 9 = 4 . 64r 10 = - 17 . 130 d 10 = 1 . 70 n 5 = 1 . 77250 ν 5 = 49 . 6r 11 = 123 . 475______________________________________variable focal lengthseparation 36 . 19 64 . 02 102 . 00______________________________________d 2 10 . 93 10 . 64 9 . 92d 5 1 . 50 1 . 79 2 . 51d 7 17 . 10 6 . 52 0 . 82______________________________________ r 1 is aspherical . a = 0 b = - 3 . 6110 × 10 . sup .- 5 c = - 1 . 98299 × 10 . sup .- 7 d = - 1 . 29356 × 10 . sup .- 9 e = 0 r 5 is aspherical . a = 0 b = 3 . 30453 × 10 . sup .- 5 c = 5 . 69713 × 10 . sup .- 8 d = 0 , e = 0 r 9 is aspherical . a = 0 b = - 2 . 66865 × 10 . sup .- 5 c = 1 . 00677 × 10 . sup .- 7 d = - 1 . 15295 × 10 . sup .- 9 e = - 4 . 13689 × 10 . sup .- 12______________________________________ table 3______________________________________ example 7 example 8______________________________________ | d . sub . 3w - d . sub . 3t | 16 . 57 16 . 283 | d . sub . 2w - d . sub . 2t | 3 . 01 3 . 033 | d . sub . 1t - d . sub . 1w | 3 . 01 3 . 03______________________________________ the above - described numerical examples have achieved a zoom ratio of 3 and a good correction of aberrations as shown in fig4 ( a ), 4 ( b ) and 4 ( c ) through fig6 ( a ), 6 ( b ) and 6 ( c ), fig1 ( a ), 10 ( b ) and 10 ( c ) through fig1 ( a ), 12 ( b ) and 12 ( c ), fig1 ( a ), 15 ( b ) and 15 ( c ), and fig1 ( a ), 16 ( b ) and 16 ( c ) . incidentally , fig4 ( a ), 5 ( a ), 6 ( a ), 10 ( a ), 11 ( a ), 12 ( a ), 15 ( a ) and 16 ( a ) show the aberration curves in the wide - angle end ; fig4 ( b ), 5 ( b ), 6 ( b ), 10 ( b ), 11 ( b ), 12 ( b ), 15 ( b ) and 16 ( b ) in an intermediate position ; fig4 ( c ), 5 ( c ), 6 ( c ), 10 ( c ), 11 ( c ), 12 ( c ), 15 ( c ) and 16 ( c ) in the telephoto end . the necessary total number of constituent lens elements is very few in view of the zoom ratio .