as applied to Micrometric Observations. 443 



from S at the diaphragm, the secondary wave due to the 

 disturbance at Q would have to travel along a path QRTP in 

 order to reach a point P on the screen, being regularly 

 refracted. 



But since P f is the geometrical image of P, all rays which 

 converge to P (i. e. pass through P) after refraction, must 

 have passed through P' before refraction, to the order of our 

 approximation. 



Hence the ray through Q which is to reach P must be 



P'Q. 



Moreover, P and P' being conjugate images the change of 

 phase of a wave travelling from P' to P is constant to the first 

 approximation and independent of the position of Q. 



Now the disturbance at P due to an element dx dy of the 

 diaphragm at Q is of the form 



^^ sin 2^ -SQ-QR-m. BT-TP), 



where X is the wave-length, t is the period, A is a constant, 

 (i is the index of refraction of the material of the lens, and h 

 is put instead of QP outside the trigonometrical term, because 

 the distance of the lens from the diaphragm and the inclination 

 of the rays are supposed small. 



But P'Q + QR + fiUT + TP = constant for P. 



Therefore the disturbance 



But 



= A^ sin ^^ -SQ + P'Q -const.) 



SQ 2 =(.z-U) 2 + G/-V)*-r-W 3 



P # Q , =(*+^) f +(y+^) i +/ s - 



Now in practice x, y, p, q are small compared with b\ b, f, 

 or W ; U and V are small compared with W. Neglecting 

 terms of order IP/W 3 , xW/W, & c ., we find 



In like manner 



FQ-Z/a ft— 4- ^ + ? 2 4-^4- W 4- L f!±ff! 



remembering that / is very nearly equal to V because the 

 diaphragm is very close to the lens. 



