376 MATHEMATICAL APPENDICES 12* 



In addition to this we obtain from the four other equations the 

 simple conditions 



= uj^ + v^Vi + w l hu l + . . . 4- UXCU N + V N GV N + W N CW K , 

 = ht,! +....+ CU N 

 = &?! + . . . -f Sv N 



= llU l + . . . + faVy 



to which the variations are subject. The variations are therefore 

 not perfectly arbitrary magnitudes, but are in such wise dependent 

 on each other that four of them are determined by the remaining 

 3N 4. The values of these last 3N 4 are limited only by the 

 condition that they must be infinitely small : for the rest, how- 

 ever, it remains perfectly arbitrary what values we assign to the 

 variations, and what ratios we take between their values. If, 

 therefore, by means of the last four equations we eliminate 

 from the principal equation 



_F l + i + +*.+ +*! 

 F t F* F. F s 



four of the 3N variations, we obtain a formula which we can so 

 arrange that its 3N 4 terms contain each a factor lu t $v, 3w 

 which may have any value whatever. The formula therefore 

 breaks up into 3N 4 independent equations which do not con- 

 tain the variations, but only the function F and its arguments. 



This elimination is most easily performed by the help of 

 initially undetermined coefficients by which the equations of con- 

 dition are multiplied before being added to the principal equation : 

 these coefficients, which I will take as 2&w, %kma, 2&m/3, 

 2&my, are then so determined that four variations out of the 

 whole disappear ; then, by reason of the 3N 4 other variations 

 being quite arbitrary, their factors are also zero. We thus obtain 

 3N equations of the form 



in which for n are to be taken all the integers from 1 to N. In all 

 these equations the four magnitudes k, a, ft, y have each one and 



