Elements of natural philosophy (Volume 2-3) . so, if the points of the image be indicated by positive whether thevalues for/, the image will be virtual; if by nega-^r^stive values, real. For a concave lens, Fu is positive,and Equation (51), answers to this case. For a convexlens, Fu is negative, and Equation (51), becomes 1-3 226 NATURAL PHILOSOPHY. Image will boreal for a convexlens as long asthe object isbeyond theprincipal focus; / = F. n F (54) // / cos 6 and the image will always be real as long as F, f u. cos 6 < 1, or COS0 Illustration. Fig. 40. That is, if-from the optical centre, w
Elements of natural philosophy (Volume 2-3) . so, if the points of the image be indicated by positive whether thevalues for/, the image will be virtual; if by nega-^r^stive values, real. For a concave lens, Fu is positive,and Equation (51), answers to this case. For a convexlens, Fu is negative, and Equation (51), becomes 1-3 226 NATURAL PHILOSOPHY. Image will boreal for a convexlens as long asthe object isbeyond theprincipal focus; / = F. n F (54) // / cos 6 and the image will always be real as long as F, f u. cos 6 < 1, or COS0 Illustration. Fig. 40. That is, if-from the optical centre, with a radius equalto the principal focal distance, we describe the arc of acircle, and this arc cut the object, the image of all thatpart of the object in-cluded between thepoints of intersectionA and A! will be vir-tual, while that of theparts without these lim-its will be real; if thedistance of the objectexceed that of the prin-cipal focus, the wholeimaore will be real. § 63. Multiplying both members of Equation (51), bysin 0, it becomes. Equation (51)transformed; ,,, . tan 6 J . (55) COS $ and giving to 0, its greatest value for any assumed object,f tan 0 will be the length of that portion of the object on ELEMENTS OF OPTICS. 227 the positive side of the axis as long as & is positive and less Explanation ofthan 90° ; / sin d, is the distance of the extreme limit 0fterms;the image of this portion of the object from the axis; andwriting f tan 0 = #, Substitutions; /sinO = S„, Equation (55) becomes, after dividing both members by/tan d, • Equation (55) J , 77T transformed; COS & If the linear dimensions of the object be small as com-pared with its distance from the optical centre, we may compared with sensibly coincide with £/y5 and the above equation °P ] write unity for cos 0, the image will, § 48, and Eq. (52),itsdistancefromsensibly (reduces to // — // f+K (56). In which the essential signs of all the quantities correspondto a concave lens. For a
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