286 DEVELOPMENT of a certain 



tions fliall fucceflively have place, it appears, that we can aflign 

 an indefinite number of elliptic arches, reckoned from the ex- 

 tremity of the conjugate axis, fuch, that their. relation to the 

 whole elliptic quadrant fhall be expreffible by a finite equa- 

 tion. But we have feen (Art. 13.) that, correfponding to each 

 of thefe arches, there is another, reckoned from the extremity of 

 the tranfverfe axis, fuch, that their difference may be exprefTed by 

 a certain ftraight line ; therefore, it appears that we can aflign in- 

 numerable arches, reckoned from the extremity of either axis, 

 pofTefling each the remarkable property of having their rela- 

 tion to the whole elliptic quadrant expreflible by a finite alge- 

 braic equation *; 



16. For example, let us fuppofe cof i<p zz — e 1 , and therefore 

 fin 2 <p := ^JJL. — _i- then, fince fin 2<p' — 1, and fin 4^, fin 8<p"', 

 &c. are each =. o, we have eK — e (* + * = _L_. __ *zz£. now 



we have alfo 6 zz 7, therefore zzz--{- -p^ ; hence we derive 



the following very curious proportion : 



Let the femi axes of an ellipfe be 1 and c ; take a 

 ftraight line in the tranfverfe axis, from the centre to- 

 wards either vertex, — ■■ — - , and at the extremity of that line, 



between the centre and vertex, draw an ordinate to the tranf- 

 verfe 



* Hence, by the way, it appears, that inftead of the femUperimeters of two ellipfes 

 which we have ufed in the preceding paper, for expreffing the coefficients of the 

 development of (a 1 + b* — 2a6co{p)n, we may fubftitute any two of an indefi- 

 nite number of elliptic arches, and certain algebraic functions of the axes of thefe 

 ellipfes ; therefore, the different infinite feries, which may be ufed to exprefs the 

 coefficients A and B, are really innumerable. 



