(om) 



165 



^--d^- («-«o) + -^.(6-«o) ^^•a^^^Taii^r —^y. ^;;;!y~ 



</x'o dy'o _ dx'o di/o 

 dao dbo db^ dag 



(TO)) 



in which w + w' >1 ; and by these equations wc can change the developments (T'") [68] into 

 series of the form 



_ da_ , _^ da d^a x'- _^_ d^a 



" ~ dx- ' "^ - dVi ' '^ dx'^ ' 2 ~ dx'.dV'i 

 ,_ db , . db . , d^ oT- _^ ^A 



dx' ' ~ dVi ' ^ ^ dx^ ' 2 ~ rfx'.rfv? 



(Q")) 



which contain no negative powers of any variable quantity, and which we are going to apply to 

 the solution of the succeeding problems. 



(III.) We must be more brief in the discussion of these remaining problems, namely to 

 transform the integral expressions of the preceding paragraph, and to effect the double integra- 

 tions within the limits of the present. Applying to the series (QW) the geometrical and optical 

 reasonings of [68], we find for the quantities a(*', SW, QW, which were there represented by 

 by the developments (D""), (E""), (F'"), the following transformed expressions : 



in which 5 is, as in (MP>), the distance of any assigned point x', y' upon the plane of aberration 

 from the section of the caustic surface, measured in a direction parallel to the normal of that 

 curve ; and X), S, Q, are rational and integer functions of x' and 3, or of x and y, which when 

 those variables vanish, that is at the focus of the given ray, reduce themselves to the following 

 values : 



Z)= ^^^il-f ; S=ii^.; Q = ^1l1^; (S(7)) 



y^C ?V|C zV^c 



■^('''> ?«) ?zi *> ^i having the same meanings as in [68.] If then we integrate the two last of 

 these expressions (RP') within the double limits afforded by the equations 



3 = 0, x'2 + .y^ = r", (TW) 



of which the former represents the caustic section, and the latter the circular circumference, we 

 shall have the required expressions for the quantities that we denoted by S^'^^ Q^'), at the be- 

 ginning of the present paragraph. 



