190 Mr. Gompertz on Series which may he summed 
and therefore every time z becomes, by flowing, any multiple 
of 360° ; K being introduced, it will introduce an additional 
correction, which will remain afterwards. 
If this equation be multiplied by z and the fluent be again 
taken, we shall have + + &c 
~ — fluent of [sine of k . z log. of — -■ 4 - O — — . z 
cos. of Q which fluent is easily found by infinite series, but 
if k be = o we shall have cos. of k = 1, and the fluent = Q z 
— —■ independent of the correction, that is cos. of z -f- c ° s ‘ ** 
4 - ----- p &c. = A — Qz -f- A standing for the correction : 
if z be = to the arc of 90° or Q, we shall have cos. of z= o, 
cos. of qz= — 1, cos. of 3%=o, cos. of 4,z= 1 &c. therefore 
we shall have by substitution — A 4. J. — & c - = A 
+ = A — — Q 1 , but if in the equation z be taken 
= 180 0 or qQ, we shall have cos. of z = — 1, cos. of 2 z = 
«|- 1, cos. of gz = — 1, &C. &C. .'. — 1 + ~l — -p 4 - 
= A — 2Q l 4- Q l or A — Q*, which series being the same as 
the other series when multiplied by 4, we have A — Q l = 4 A 
-3QV.A = f Q\-. 1- 
, r , COS. of 2Z , 
and cos. of z 4- — ■ — 4“ 
J_ 4. i. „ See. 
z l 3 * 
COS. of 33 
"”9" 
Q-A= 
Q 2 
&c. 
fQ*-C* + -7. 
It is remarkable that this equation is true, not only when the 
equation from which it is derived is true, but likewise when 
2=oor 360° in which that fails, and that the correction might 
have been sought in those cases had this circumstance been 
known. Multiply this again by z } and find the fluent, and we 
