254) Mr. J. Gordon on Dr. Lardner's and M. Lacroix's 



And as I am not aware that there is a satisfactory demon- 

 stration of either of these theorems in our language, I have 

 annexed one, which, excepting some alterations, is the same 

 as that given in the quarto edition of Lacroix, on the Differ- 

 ential and Integral Calculus, vol. i. Art. 107. 



I have the honour to be, &c. 



Public Commercial and Mathematical JAMES GORDON. 



School, Aberdeen, 



1st. Remarks on the Demonstrations of the Theorems of 

 Lagrange and Laplace as given in the above-men- 

 tioned Treatises. 



Using the same notation as in the note to Lacroix: 



v, V s , v\ &c. (page 637) are there put = p, p\ p 5 , &c. 



x <2. r 

 when* = 0. Now although < q^ + 2 qip^ I becomes 



x* f 1 



y-2 fylf + 2 yi&P when * = 0; yet, since v= 



+ Sec., will not the coefficient of that term of v which contains 

 the first power of #, go to the formation of the coefficient of 



n/j& 



- -cT~o an d not vanish? but, by making v actually = p, this 



part of the coefficient will be lost. A similar observation ap- 

 plies to the other coefficients. 



If I am correct in this remark, the demonstration of La- 

 grange's theorem, beginning at that part which is near the 

 bottom of page 636, will be as follows. 



Consequently, 



+ &c. 

 But q =f(z) 



And, 



