OF MULTIPLE ARCS. 287 
always subsists. Hence ¢,=1, ¢,=1—¢?, &c. The law of the for- 
mation of series (3), expressed in equation (4), being the same with 
that of series (1), expressed in equation (2), the difference between 
the series (3) and the series (1) arises solely from the difference in 
their first terms. The general terms T,, and ¢,, are easily found. In 
fact, m being any number aes than zero, 
T,,== 1—mt? +o -3)— = ee NA ee 
(m—5)—&e., BS eee ua (5) 
(m—3)(m—A4) - (m—4)(m—5)(m—6)Eé6 
Eines 
Hei Oe Cosreret Watiich ie ibauied cece CO) 
m-+- 2, 
2 
and ¢,,=1—(m—2)¢?2 + 
When m is even, the number of terms in the value of T,, is 
and > in the value of #,,. When m is odd, the number of terms in 
each of the expressions T,, and ¢,, is 
une H To prove (5), we observe 
that T,=1, and T,=1—2¢?. Henee the law is true for the first 
two steps. Assume it to hold for m—1 steps. Then 
(m—1)é* 
,2_ 
and 727 Eo = ¢? —(m—2)t*+ &c. Therefore, by (2), 
Ty =1—(m—1)02 + =” (m—3) tee. 
4 
yee = (Ga ibe 
which proves the Law universally. In the very same manner equa- 
tion (6) can be shewn to hold. 
§2. The following formule may now be established : 
If 2¢ cos €=1, and 2¢ sin 6=A, 
theni2¢™1cos:mO— Wait yee chs ee cevees (CP) 
BNE) 2 APES iO — Meh alah (aac ot catiooinns saan) 
§3. To prove (7), we remark, that, by hypothesis, the Law holds 
for the first step, that is, when m=]. Assume it to hold for m—l 
steps. We have only to shew then that it holds for the succeeding 
step. Now, since the Law holds for m—1 steps, 
