of certain differential Expressions , &c. 273 
By the method given in the preceding pages, the coefficients 
are made to depend on the integrals (/, F) of dx J )» 
— . These integrals, it is necessary to compute ; and 
dx 
methods have been given for that purpose for all values of e, and 
consequently for all values of a and b. If the coefficients are to 
be determined by deriving A, B, from A', B', &c. the best method 
to be followed, is that given by Mr. Ivory, who determines the 
coefficients, when the index — 1, in fact, by integrating 
° r ° n WhicU A dependS ’ 
and dx J f J ~L*» ) (df), or S)“, on which B partly 
depends.* 
The author last mentioned, in his valuable Paper inserted in the 
Edinb. Transactions, first, I believe, applied the method of trans- 
forming /, F, into similar integrals f, F', /", F", &c. to the 
determination of the coefficients A, B, See. ; but the method of trans- 
formation belongs to Lagrange. This great mathematician has 
also solved the problem of the expansion of (a 2 + & 2 — 2 ab. cos. ; 
he determines A and B, when the index ~ n + *- = a, in which case, 
the series for A and B, with respect to its numerical coefficients, 
decreases the fastest. But the solution is not general, or, to speak 
* B depends on/ and F, for 
cos. 0 . d 9 (zx z —i) 2 dx 
V(i— e a . f cos. — 
+ 2. 
(2 — e 
—4 (1 — e z x 2 ) dx 
a ~ V(i— x 1 ) (i — e? x z ) ~~ e z ' VC 1 — dc 7 -) (i— e z x z ) 
—e z ) dx 
■) gf -4, / , 2 .( 2 — 6 1 ) d 
(i-e z x z ) - e z W \ I- X *r 
'X z )(i —s z x z )' 
+ I assert this on the authority of Lacroix, having never been able to procure the 
volume of the Turin Memoirs in which Lagrange’s method is contained. 
N n 2 
