The cosine-fourth-power law states that the irra dia nce 1 a t a n y p oin t in the image formed by a photographic lens, in the absence of vig netting , is equal t o Eo cos' {3, where E o is the irra diance at the center o f the fi eld , and {3 is the a ngle between the axis of the lens and the con juga te chief ray in the object space. Althoug h not us ua ll y so st a ted , this law in volve s th e addition a l ass umpt ion that the lens is free from dist ortion. \,yit h this a ss umpt ion , t he la w applies rigorous ly when t he dia phragm is between the lens a nd object a nd the object is a t an infinite d ist a nce. If the diaphragm is betwee n t he lens a nd the image plane, t here m ay be cases in which the irra di a nce fa lls o ff les ra pid ly f rom t he center of the fi eld outward than is p redicted by t he cos in e-fourth -po\\'er la w. A t ype of negative di st ort ion is d efin ed for wh ich the irrad ia nce is un ifo rm o ve r t he entire image. When t he di aphragm is within t he lens syste m (t he m ore common co ndi t io n for photog raphic le nses) , o ne mus t k now t he dist ortion of the po rtion of the lens fo ll o wing the di aph rag m before a d efi nite st atem e nt regardin g the irrad ian ce of the image ean be ma de. The departures from exactness of the cos in e-fourth-powe r la w a rise p a rt ly because t he efl"ectiv e a rea of the entrance pupil is a fun ction of t he obliquity of the incident chi ef ray. A m ethod is given for mcaRuri ng this vari a tion in effecti ve a rea.
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I. C. Gardner (1947) studied this question.