by a Method of Differences. 175 
t'+zr—idv+t —r_v _j_ or — t0 taken according as r is odd 
or even, that is, f -\-r— i.tv-\- ^ if r he odd, and — -rtv — 
T -^d. m if even. And thus we might proceed to the discovery 
of an infinite variety of theorems relative to the sines and 
cosines contained between any two limits in a circle, and th<* 
consequent inferences, the method being capable of a very 
extensive application ; but rather than tire the reader’s pa- 
tience with what he may effect himself from what has been 
already said, if there should otherwise have been any diffi- 
culty, I shall propose 
Theorem V. 
If we have the sum of the series, a sine of pz-\-b sine of 
p-fq.z - fc sine of p-\- sg.z-f d sine of p-\-§q.z Sec. expressed 
generally in terms of p , q, and z, we may find the sum of the 
series, a cos. of pz~\-b cos. of p-\-q.z-\-c cos. of p-\-v,q.z +d 
cos. of p\ 2 >q- % & c * expressed generally in terms of p, q, and 
z, and the contrary. 
For if we put 90 °-\-pz for pz in the series, and in the ex- 
pression for its sum, we shall have instead of the sum of the 
series, a sine o f pz-\-b sine of p+q.z-\-c sine of p-\-<-iq.z &c., 
the sum of the series, a. sine of 9o°-J-^ 2: +fr- s i neo f 90 °-\-p-\-q.% 
- {-c sine of go°-\-p-\-2,q .% &c. or because the sine of an arc is 
equal to the sine of 180° — that arc, we shall have the sum of 
the series a. sine of 90° — pz-\-b. sine of 90° — p-\-q.z See. or its 
equal, a . cos. of pz-\-b . cos. of p-\-q.z-\-c . cos. of p -f- zq .z Sec. 
which is the first part of the theorem ; and by following the 
