by a Method of Differences. 
393 
— g sin ----- i > — q ' % t.~E e oi 4 if this should have a failing case, 
it will be, by this scholium, when qz = o or some multiple of 
360° provided g be affirmative ; but if g be negative, it will be 
when qz is some odd multiple of 180° ; a similar expression to 
this is given by Mr. Landen by his method. 
If p be = q = 1 we shall have, sine of z -\-g sine of zz -j-g* 
■, if we now multiply by z 
sine of « 
sine of <\z &c. = -, , , , 
o g + 1 — 2 g cos. or z’ 
calling the cosine of %, x and find the fluent, we shall have 
cos. of z + - — - 4 T— &C. = - . log. of i-f -g -2 gx 
J o 
which has no failing case. 
6 . According to this General Scholium, Example 2. to Theo- 
rem II. has failing cases in the investigation, unless the series 
1, r, r. &c. converge; thus those in the Corollaries 1. ir. 
and hi. when qz is any odd multiple of 180°. By bringing 
both series to one side in the equations in Cor. 11. we have 
sine of qr— p.z -\- sine ofpz — rxsine ofr/.r-j-i — y>.%-|-sine of p-\-q.z 
4 - &c. = o, and cos. of qr — p . z — cos. of pz — r x cos. of 
q.r+l—p.z — cos. of p -j- q ■ z + & c - = °- Multiply them 
both by z and take the correct fluents when % — o, and we get 
cos. of qr — p . z , cos, of pz 
qr-p 
4 ” 
— r. 
cos. of q ■ r+ i — p . z . cos. of p + q . z 
q r 1 — pz 
■4 
r . 
r + 1 cos. of q . r 2 — p 
2 q . r -J- 2 — pz 
+ 
cos. oI r p -f 2 q . z 
P + 4 
&C. 
P+q 
M, and 
+ 
sine of qr — p.z sine of pz sine of q -j. \ — sine of q z 
V — P P q . r -f 1 — p p -f q 
-J- &c. = o, M being put for ~ — - 4- r. 
4 r. 
r+i 
2 
q ■ r - (- 2 — p 
MDCCCVI. 
4i 
p + zq 
qr-p 
&C. 
4r 
qr 
q • r + 1 — p ’ P+q 
r + 2 . q 
■r. 
p.qr-p 
C c 
p+q.q . r+l-p 
