by a Method of Differences, 
167 
s= — sine of pz + r sine of p -f- q . z — >, &c. and from cos. of 
p% — r cos. of/> -j- a . z, &c. = jJ L. by the like sub- 
*■ * 1 2 cos. of 
stitution we get cos. of qr-~ p.z—r cos. of r-f 1 . q ~p . % + r 
— cos. of r+a .q— p.zSzc. ■ 
cos. of ±qr — p.z cos. of p — \qr.z 
2 cos. of £<p| r z cos. of \qz\ 
cos. of pz — r cos. of /> 4~ <7 • ~ 4“ r ’ cos * °f ^ ^ 
and these by Cor. 1. are true, whatever r may be. 
Cor. hi. If p be = j - qr we shall have, cos. of -§- qrz — r cos 
of 4 . r+2 . 2 + r. — . cos. of ~q . r -j- 4 . £ &c, = 
cos. of o 
2 cos. of \q«\ 
— _ ... 1 - ) or if A be written for we shall have, cos. 
2 cos. of \qz\ 
of rA — - r cos. of r -f- 2 A -J- r • cos « °f r + 4 • A — & c * 
= - ■ * — ■ ; which is the same in substance as Simpson's 
2 cos. of A| r 
lemma, page 67 of his Tracts. 
Cor. iv. If we put pz = 180 0 — - tz, qz — 180 0 -— sz we shall 
have according to Scholium in. at the end of the examples to 
Theorem I. 
sine of tz -j- r sine of t -f- 5 . z -f- r - ~~ sine °f t 2 s . z &c. 
sineofi8o° — tz — Ir.iSo— sz sine of go 0 . r + t— ^rs z i n 
== — ■ ■ 2 — - ■ — and — cos. of 
2 cos. of 90° — A qz\ 2 sine of \ sap 
t z — r cos. of £ -j- 5 • s — - r , — -■ cos. of £ -f* 25 * ~ & c - = 
cos. of 180 0 — tz — \r . 180° — 
sz 
2 cos. of 90° — | 
cos. of 90°4-f_irs . z r , , 
2 =rr- .*. cos. of tz -f r 
2 of sine \sz\ 
cos. of £ -J- s . z -j- r. iihi cos. of £ -J- 2s. 2 &c. = — -- -°- f 9 ° r+i ' rs - z 
2 2 sine of ± sz| r 
and if in these r be a whole number, and p and q be written 
