192 PHILOSOPHICAL fRANSACTlONS. [anNO 1787. 



X v/— 1= +L+M\/ — 1, in which case the two serieses + l and + m 

 converge, and {T -{- A \/— l) x(±l + m^— l) will be a value or root of 

 the given quantity. 



In the same manner the remaining roots may be deduced. 



2. Let ± p be — p, multiply — p ± q -/ — l into — 1, and it becomes p 

 + a -v/ — 1, a quantity of the same formula as the preceding; let r + ^' v^ — 1 



be a root of the equation x + 1 = O5 then will (r' — A'-/—]) (+ l + m 



i/ — = ± h' ± k' -/ — 1, be a root of the given quantity. Otherwise; the 



root may be deduced from the above-mentioned series by substituting in it for 

 I I I 



— (p)" its value p'" X (— l)", and it will become the same as the preceding. 



I 



3. Let p be less than q, and the value of(+p±av'— l)'^ may be deduced 



from the preceding series by substituting in it ± a -/ — 1 for p, and + p for q. 



I I 



Otherwise, since (±p± aV— 1)'^= ± %/— I X (a + py'— l)*^, and the 



root of (a + p v^— l)'- can be deduced by the preceding method, which suppose 

 l' + m' -/— 1; multiply this root into h ± V' — i, where h -j- © y' — 1 

 denotes a value of the root + ^ — i, and the quantity resulting will be one 

 value of the given quantity. The remaining values can be deduced by the same 

 method. In this case the given quantity is resolved into a series ascending ac- 

 cording to the dimensions of p, and descending according to the dimensions of 

 a; in the former case it was resolved into a series ascending according to the 

 dimensions of q, and descending according to the dimensions of p; both the 

 serieses affording the possible or impossible parts will always converge. 



4. If P = + a, then will (± p ± p /— l)^ = p" X (± 1 ± /-- l)^ = p^ 



^ - - 1 



X 2^(± /4- ± ^ — 4-y=P'' X 2^'^ X V— 1; ^or ^ — 1 = + ^/±± ^ — x. 



4. 2. When p = 0, or a = O, then it becomes the first case y' ± a. 



5. Let p = a + a, where a has a very small ratio to q; then will (p ± a 



- A 1 11 



(p ± (p ± x) >v/— ly = (p X 2^ X (— 1)* ± a /— 1)^ = P'^ X 



I — r I — r 4r 



ixP~X2— x'-^(-l)X /(- 1)^-1 xi^'x 



'-:/(-l)X^^±7.V^.^-^'XP^'\2 - X 



I— Sr, 



(— 1) X ^ {— 1)*^+ &c. In this series the same root of the quantity V~"l 

 is always to be used. 



6. If in the given quantity are contained more quantities of the above-men- 

 tioned kind or their roots ; then, by repeating the same operation, can be de- 

 duced the roots or values of the given quantity. In some cases the impossible 

 part may vanish, which may be the case in a quantity of the following formula. 



