442 PHILOSOPHICAL TRANSACTIONS. [anNO I76O. 



Hence, writing a for one 4th of the periphery of the circle whose radius is 1, 

 and taking x equal to the said radius, we find hyp. log. -7= = — ~r ; and con- 



■V —I V — J 



sequently hyp. log. V~i = -t-=, and hyp. log. •— 1 = + 



2. The hyp. log. of ~^ being = ^ 4. |' + ^' 4. i*, &c. 



p' = fluent of i hyp. log. ^-, is -^ ^ + ^i + f! + i^, &c. 



p" = fluentof i P'=^+ I +^, + i;, &c. 



p'" = fluentof ^P" = :r + |;+|; + |:, &c. 



P*' = fluent of f P''' = ^ + 1; 4. 1; -j- ^^, &c. 

 &c. &c. &c. 



3. By writing, in the first equation in the preceding article, - instead of x, 

 we have 



Hyp. log. -L- = a^-> + ^ 4. ~, &c. 



X 

 1 ^1 



But the hyp. log. of ^ is = hyp. log. j---j = hyp. log. — - -f hyp. log. x 



X 



+ hyp. log. — 1 = + 2/; + X 4- hyp. log. ^— , b being put for ~^, and x 



for the hyp. log. of x. It is evident, therefore, that 



Hyp. log. ^— is = -I- 2^» — X -|- ar~' + — -f — , &c. where, of the two 



signs prefixed to lh, the upper one takes place, when the hyp. log. of — 1 is 



_ 2 a . . . 



taken equal to . — , likewise when x is taken equal to -v/ — 1 ; and the lower one 



takes place, when the hyp. log. of — 1 is taken equal to —t-=, also when x is 



V — 1 



taken equal to , — : therefore, if we observe to take the value of hyp. log. of 



— 1, as last mentioned, and a: equal to-j=, instead of -/ITT, we need retain 



only the lower of the said signs. 



4. For brevity sake, we shall, in what follows, put the series 



1 + ^, + 3-, + i^ &c. = P", 



1 + <4 + .^ + A, &c. = P'^ 



3* ^ 4*' 



&c. &c. 



^ + 2^' + J« + l^' ^^' — ^'^ 



