108 Prof. PowelPs Abstract o/M. Cauchy's 



then, g will be the perpendicular distance of the point x y z^ 

 from a plane passing through the origin whose equation is 



ax-{-by-^c^ = 0. (16.) 



The equations (13.) will by this notation become 

 f = ^ [d cos k g + g sin ^ §] 

 >) = 5* [e cos^g + h sin A- ^] (17.) 



^ = -5* [ f cos kq -\- i smk q] 

 Now to determine the coefficients d e, &c., in functions of 

 the variable / and the arbitrary constants k ah c, we may pro- 

 ceed as follows : 



Let 5 be the angle formed by r with O P, 

 then cos 8 = a cos a -\- b cos /3 + c cos y. , (180 



Also from the values of Ax, &c. (2.) joined with that of q 

 (15.) we have, taking the corresponding small increments, 



Aq = aAx -|- bAy + cAz — r cos 8. (19.) 



Also we shall find by a simple trigonometrical process 



A cos /t' § = — 21 sm^ j cos k q 



— sin [k r cos 8) sin k q 

 And in exactly the same manner )- (20.) 



.. -7 o / • 9 ^^ ^' cos 8 \ . - 

 A sm kq = — 2 1 sm^ ~ \ sm k q 



-\- sin {k r cos 8) cos k q. 

 In order to simplify the subsequent investigation, we will 

 in the first instance consider the sums of terms (17.) as each 

 reduced to a single term, or take 



^ = d cos k q -\- g sin ^ ^, 



which on differentiating with respect to q gives 



-1 — = A: [ — d sin ^ ^ + g cos k q) ; 



and substituting these values in the corresponding formula 



A f =8 d A cos k q -\- g A s\u kq, 

 we shall find the value of that quantity, and by similar means 

 those of the others analogous to it, 



,. *. • -o { k r cos 8\ sin (kr cos ^) d^ 

 Ag = -2 f s.„^ ( -^— ) + ~^Tc HJ 



>(21.) 



