1818.] On Reversion of Series. Ill 



The equations intended are 



cos. y = sec. x :. cos. x — sec. y. 

 Equivalent to the squares of the first, w e have 1 — sin. *y = 1 

 + tan. *x whence sin. y = tan. x . a/ — 1. T he flu xions of 

 these givey cos.y, or its equivalent^ sec. x = x V — 1 . sec. *x. 

 Whencej) = x a/ — 1 . sec. x. Hence again * = — y*/ — 1 . 

 cos. x = — y v' — 1 sec. j/. Assuming y — z V — 1 these 

 last equations become 



z = x sec. x and * = % sec. z V — 1 

 which furnishes an evident example of the third case ; the only 

 one, in fact, at present known. 



If the fluents be taken, we shall find 



i • , . . ^ i 1 + 'an- h v , , 1 + sin. x . 



z =log.tan. (± * + ±x) =log. T - r[ -f-=±\og. j—^ &c. or 



* - * + * * + A "«• + sUv * 7 + rfHir x * + &c - 



as has been shown by Baron Maseres in the Scriptores Loga- 

 rit/tmici, by Professor Wallace in the Math. Repository, N. S. 

 and others. 



To elucidate the mode of generalization, I adhere to the first 

 of these formulae, as in my former communication. Expressed 

 as a function of x */ — 1 , it gives the general values 



z = (?-' log. tan. H 7T + ^== f (x V - 1) j 

 x = —L== p- log. tan. ^ n + — != <p (- z) ^ 



£x. 1 . — Make <p x = V'tia . x, then are 



z =■ x + ax 3 + £ a* x' D + uu> « 3 x 7 + y^ 7 « 4 x 9 -f &c. 



x = z — a z 3 + f a* z 5 — yj « 3 z 7 + *_y «* Z 9 — &c. 

 and by assigning different values to a, series may be formed at 

 pleasure. If a = 1 or x or A, the coefficients will be more 

 simple than in the original series. 



Ex. 2. — As a general example of a large class of series, assume 



/ 6 m a "'- 



6 x = y/ . x", TO and n being any odd numbers, and a 



optional. We shall then have 



the upper sign obtaining when, m and // l)eing divided by 4, each 

 leaves the same remainder; and the lower sign, wlien those re- 

 mainders are different. The reason < fwhich.is, that ( v / — I) 4 being 



