Mr. J. Gordon on the Expansion of Functions. 253 



both ways infinitely produced). When b = 0, the curve is 

 a circle to radius a cosec /3. 



Scholium. The following formulae for transferring the equa- 

 tion of a curve from one pole to another, are useful in these 

 and many other problems ; they have never, to my knowledge, 

 been published. 



Having given the equation of the curve to the pole A and 

 axis AX, to find its equation when referred to the pole B, and 

 axis BY; the position of A and B being given : 



Let AX, BY intersect in I, making 

 Z. XI Y = % ; and put XAP = $, 

 YBP = 9, IAB =/3, and AB = <z: 

 then the two equations may be ex- 

 pressed, functionally, thus : A P = 8($) 

 and BP = 8' (0). Draw P a, B b AX, 

 and complete the parallelogram abcP; 

 then Pa = 8(<p)sin$, A a = 8($)cos4>; 

 B b = a sin /3, Ab = a cos /3 ; P c = 8' (0) cos (0 -f x), and 

 B c = 8' (0) sin (0 + x). .'. 8 (p) sin 4> = sin /3 - 8' (0) cos 

 (0 + x) ... .-. (), and 8 (<j>) cos <p = a cos |8 8' (0) sin (0+x)- 



asin/3 8'(0) cos(0+%) rp,. 7 r 

 .-. tan <p = - ^ ^y /M A , r This value of and the 



acos/3 8'(0) sm(0+%) 



proper value of 8 (p) substituted in (a) will give 8' (0) in terms 

 of and known quantities as it ought to be. 



I am, Gentlemen, your obedient humble servant, 

 Seamen, near Scarborough, CHARLES GlLL. 



Sept. 6th, 1830. 



XLIV. Remarks on the Demonstrations of the Theorems of 

 Lagrange and Laplace, given by Dr. Lardner and M. La- 

 croix for the Expansion of 'Functions ; with a Demonstration 

 of those Theorems. By Mr. JAMES GORDON. 



To the Editors of the Philosophical Magazine and Annals. 



Gentlemen, 



T SEND you the following remarks upon the demonstra- 

 * tions of the theorems of Lagrange and Laplace, for the 

 expansion of functions, in order that some of your correspon- 

 dents who are conversant with these subjects may correct my 

 views if erroneous. 



The demonstrations referred to are contained in the two 

 treatises on the Differential and Integral Calculus, by Dr. Lard- 

 ner and M. Lacroix; in the latter the demonstration is given 

 by Mr. Peacock in one of his excellent notes to the work. 



And 



