occurring in Integrals of Partial Differential Equations. 299 

 distance from the origin of or, and leta ^ = ~; then, whatever 



Y il-z) +f{l + z) = (A) 



Hence by Taylor's Theorem, 



F(0+/(0-(f'(/)-/'(0)^~+(f"(/)+/"(0)2-&c.=o 



independently of the value of 2;. 



Therefore F (Z) + / (Z) = (1) 



F(Z) -/'(/) =0 (2) 



F" (/)+/"(/) = (3) 



&c. &c. 



These equations are to be satisfied by a consideration of the 

 forms of the functions F and f, so as Z may have in all the 

 same arbitrary value. Equations (1), (3), (5), &c. are plainly 

 satisfied by makingy the same as — F. Hence from equation 

 (A) F ( Z — ;:) = F (Z + 2), and F (Z) must be a maximum 

 or minimum. Therefore F' (Z) = 0, and equation (2) is satis- 

 fied. But we must also have F'" (Z) = 0, F" (Z) = 0, &c., that 

 is, a function of x is to be found such, that the same value of 

 a-, which makes it a maximum or minimum, makes all its odd 

 differential coefficients disappear. We are consequently led 

 to a trigonometrical function, and the simplest is sin (x + c), 

 which both satisfies the condition F (Z — z) = F (Z + 2;), and 



gives for the value of /, - — c, an arbitrary quantity. Again, 



the only other mode in which any of the equations (1), (2), 

 (3), &c. may be satisfied, so that I shall remain indeterminate, 

 is by makingy the same as F. By this supposition we satisfy 

 (2), (-t), &c. ; and equation (A) shows, by making z = 0, that 

 F (Z) = 0, so that equation (1) is satisfied. At the same time 

 F" (Z) = 0, F"' (Z) = 0, &c. Hence a function of j; is to be 

 found, in which the value of ^, which makes it = 0, makes 

 all its even differential coefficients disappear. The function 

 sin (x + d) is plainly applicable ; it satisfies the condition 

 F (Z — 2) = — F (Z + z), and gives for Z, — c', an arbitrary 

 quantity. In two ways, therefore, we are conducted to the 

 same form sin (.r + c), or, what is equivalent, sin x. Also, 

 every function which will satisfy all tlie above conditions must 

 be included in the general one sin x -f- ?h, sin '6 x -\- in' sin 

 5 X + &c., containing an unlimited number of terms. But as 

 this consists of terms of the same form as sin x, and would in- 

 tlicute a motion resulting from motions of the kind indicated 

 by sin r, we have a right to conclude that the ^>?7/ho;3/ form 

 ol" the function is sin x. If A = the interval between two con- 

 secutive points at which the curve, whose ordilialcs arc pro- 

 2 Q 2 portional 



