of certain differential Expressions, See. 275 
d 9 y/( i-|- « cos. $) and — — - q ' s fl . Now, the coefficient affecting 
co s.n 6 =af cos. nQ.d&,( 1 — c. cos. 0)~r~, (a a constant quantity,) and 
c6s.‘»fc=§-'| (2. cos. &y — n. (a.cps. 0)' , ~ 2 -h 7-^7^ (scosJ) ~ 4 — &c. j 
E, E', &c. Legendre calls ellipses, because the differential of 
the arc of an ellipse may be represented by an expression as 
d6y/{ 1 — efcos. 9)"-) ; but the problem of the expansion of 
(a* -f t — cos. $)~r~ requires only the integrals of d9. R; 
the determination of which integrals, is totally independent of 
ellipses, as it is, of all other curves. 
That the determination of the coefficients A, B, &c. depended 
on the integral of Rd9, which, in a particular application, represents 
the arc of a conic section, was known to D*alembert. In the 
Recherches sur differ ens Points importans du Sysieme du Monde , page 
66 , he proves that A, B, are respectively equal to —7-, - c be- 
ing the semicircumference of a circle, and M, M-, the integrals of 
dz (a-\-b cos. z)~, and cos. z. . dz . (a-{- b cos. z}~, when z = 
which integrals depend, he says, on the rectification of the conic 
sections; he then adds a remark, which requires some comment 
and explanation. “ Tout se reduit done a trouver par approximation, 
“ la rectification d'un arc donnd dans une section conique ; et e’est 
“ a quoi on peut parvenir aisement par diff^rentes mdthodes. Mais 
“ je ne m^tendrai pas davantage la-dessus, parce que cette maniere 
“ de trouver les inconnues A et A', me paroit plus curieuse et plus 
“ g^ometrique que commode pour le calcul.” p. 67. D'alembert, 
therefore, rejects a method which has since been adopted : the 
reason, I presume to be this ; if he had attempted to find the co- 
efficients by the rectification of the conic sections, he must have 
reduced the integrals of dz (a-^b. cos. z)~r, cos. zdz. ( a-\-b cos. z)- r 
