iQs Mr. Gompertz on Series which may he summed 
correction, the same as Landen finds, this is true whilst z is 
exclusively between o and i8o°. 
4. In Cor. 11. Example 1. Theorem I. p being =1 we have 
sine of 2 + sine of 32 4 - sine of kz & c . = — =— failing; 
(from above) when qz or 2 z = o, or a multiple of 360°, and 
therefore when % = o, or a multiple of 1 8o° ; if we multiply 
this by z and find the fluent we have, because r— ( by 
J 7 2 sine or z v J 
putting y for the sine of z ) = 
— 1 
_ y y 
v'r— 1 
y 2 _i_ 
V II— '2— 1 
V X z — 1 
{x being put for —) whose fluent is = — •§- log. of x -{- s/ x x — 1 
L+iSa 
y A . 3 
\ log. of — - , consequently cos. of 2;+ -J- 
cos. of 52: 
I log. of 
1 +^i—y* 
correction, which being sought 
s - 0 y 
when 2=90° and consequentlyy=i and the cosines of 2, of 32;, 
of 52;, &c. = o, we have it equal to o ; and this has no failing 
case since it will not fail when z = o or any multiple of 180° 
in which primitive equation does. If z be = 45 0 we shall have 
cos. of z = s/ cos. of 32: = — s/ cos. of 52 = s/ 
cos. of 72 = -J- v/ cos. of 92 = + s/ f , &c. therefore x/'J 
X : 1 — T - T + f + i - &c - 0r 
■S A x X _ JL -L. _ _ 2 4 
&c. = •■§■ log. of 
5 + V'i 8 , 16 
v'll 
3-5 
3-5 7-9 11. 13 
+ &C. = x/|i . log. of 
2 • 3 &c. 
7 9 
4 - -4 
* 11. 
13 
bs \/ i log. of \/ 2 -f* 1 T~, 
1| 2 & * *3-5 
~ log. of s/ 2 -f 1. 
5. By Theorem I. Example 6, we have the sum of the series 
sine of pz 4-g sine of p ^ q.z^- g* sine of p -j- M- % + &c. 
