TRACT 2S'.

and infinite serifs.

IDS

where the signs of the former series are found by changingthe signs of every other pair of terms in the latter ; namely,omitting, the first term, change the signs of the 2d and 3d■terms, then passing over the 4th and 5th terms, change thesigns of the 6th and 7th; and so on. For, by Art. 86, the

former of these series is equal to + ; and, by Art. 72, thelatter is equal to ^2.

90. Let us now consider the cases in which c 2 is greaterthan 6% which include all the cases not comprehended bythe former, or in which £ z is not greater than b z . And this,it is evident, will happe.a both when a is positive, and whennegative; namely when a is any positive quantity what-ever, or when it is any negative quantity, and a 3 greaterthan 2 b z . And in these two classes, r 2 will be positive ornegative, according as a is positive or negative.

91. Now the series in this class will be found the sameway as in the last, by only writing here the letter c beforethe letter b ; for then we shall have s — $/{c + b), and d ~4/(-c + b)~ — 3 /(c-b),

Then s =%/{c + b) =z s /c x : 1 + — —

242 i 2 . 563

3.6c 2 3.6.9c3

&C,

and d = -Z/(c-b)=%/cx 1+^ +

+

2.563

2 b

Hence s + d = — x

1 2.56*

S + 3.6.9c s

+

3.6c 2 ’ 3.6.9c3

2.5.8. 1164

3/c a " ’ 3 ‘ 3.6.9c s ' 3,6 . 9 . 12 . ISc 4the 1st root, and which was given by Clairaut . And■9 + d '1 r -6 1 2.56* 2.5.8.1164

X = 3+37779? +

&C.

&c

V'-3 I

±vSC.y/-3 x :!■

3.6.9. 12 . 15c42.5.86 4

&e

26*

3.6c 2 3.6.9. 12c 4

&CC,

for the other two roots, which are new.

92. Here it again appears, that when c 2 is positive, thetwo latter roots are imaginary; because then 3 /c x \/ — 3will be imaginary. But if c 2 be negative, those roots willbe both real; since l/c x s / - 3 then becomes %/{c V — 1) x— 3 =%/cx — — V—3 ~~%/cx */ 3. The signs pre-

fixed to the terms as above,, take place when c z is positive;but when c 2 shall be negative, the signs of the terms con-