the Connexion between two Physical Magnitudes. 443 



if h 4 be neglected. Thus 



'-'-SS m 



(A) and (B) give the analytical solution of the problem ; but 

 for practical purposes the following interpretation is important : 



o ^ f d V i d*ytf~\ . f rfy _ d 2 yA 2 \ 



so that 



In passing from the observed to the true curve, the curvature 

 is everywhere to be increased instead of diminished, and 



pm=Hm-\-±R S. 



It may be remarked that while it is always possible to pass 

 accurately from the true curve to that derived from it with any 

 prescribed range, the inverse problem is not determinate unless 

 it be understood that the range is small, so that its fourth 

 power may be neglected. The practical utility of the solution 

 obtained is scarcely affected by this consideration ; indeed it is 

 only when the curvature of the curve is considerable that the 

 correction itself is of much importance. 



It often happens that the connexion between the two curves 

 is not so simple, at least at first sight. Suppose, for example, 

 that in the case taken as an illustration the spectrum is impure 

 from the sensible width (2k) of the image of the slit. The ob- 

 served curve is then connected with true by a double integration, 



Now 



**i 



i c x+k r c* +h i 



1 C x + K K 2 d*y 



*+* m dry _ IP/fry k* dS[\ _h?d*y 

 _ 6 fa? 6W + 6 dx 4 ) ~~ 6 </> 



y be ne$ 



y'=y+ 



if the term in hrK 2 may be neglected. Thus 



6 dx* 



The rules remain just as before, except that instead of h we 



