IN GEOMETRY, MECHANICS, AND ASTRONOMY. 423 



or, putting 2f/= F, and IDu.U^ W, for brevity, 

 R= V, Dr.R = IF, 



and therefore, Dr . V = fV (35). 



Which equation indicates that F and IF are at right angles. 



V is evidently the resultant of all the forces, supposing them transferred to the origin in their 

 proper directions ; and IF is the axis of the resultant of the couples introduced by transferring the 

 forces ; for Du . U is evidently the axis of the couple consisting of U acting at u, and - U acting 

 at the origin ; and therefore 'S.Du . U is the sum of the axes of all such couples, and therefore 

 the axis of the resultant couple. Hence the condition of the forces having a single resultant is, 

 that the resultant force { V) shall be at right angles to the axis ( IF) of the resultant couple. 



This condition is simply expressed by the equation, 



= AF.IF, 

 which is got immediately by performing the operation AF on (35). See Article (13). 



25. If we transfer the forces U, U',-U", &c. to any point v, instead of the origin, the 

 resultant couple will be 2 Z? (m - u) . f7 instead of^Du.U. Now 'S.D{u - v) . U =^Dii . U 

 - Dv .^U = IF - Dv . V. Hence, if we assume (F to denote the resultant couple when the forces 

 are transferred to v, we have 



fV = JV - Dv . V. 



We may determine the minimum numerical value of IF^ as follows : 



Let X F be the projection of the line IF on the line F; then IF- \Fis perpendicular to F, and 

 is therefore expressed by a symbol of the form Dv' . F, where «' denotes a line which we do not 

 require to know. 



Hence, we have IF = \F+Z)(''. F, and therefore 



IF =XF + Z)(«'-t,). F. 



Since v is arbitrary, D(v' — v). F denotes any line whatever at right angles to F: hence the 

 numerical value of IF is least when D{v' — v) . F = 0; and therefore IF = \ F. To determine X, 

 since IF — X F is at right angles to F, we have 



AF. IF 

 AF.(IF-XF) = 0, and .-. X = ^|^-. 



Hence the axis of the couple of minimum moment is 



AF. IF 

 AF. F ■ ■ 



We may observe that the equation W^=XV indicates that the axis of the couple of minimum 

 moment ( TF) is parallel to the resultant force (F). 



These instances suffice to shew the application of the notation Du'.u, and Am', m to Statics. 



Application of the Notation Du'.u and Au'.u to the Calculation of the Motion oj 

 a Rigid Body about its Centre of' Gravity. 



•la. Let 7/ be the symbol of the position of any particle (^m) of a rigid body at any time (<), 



cPu 

 and U the accelerating force which acts upon ^m: then, since ^»» -j-j is evidently the symbol of 



the effective force on cm, the forces U^m, and - - Sm applied to Sm, and similar forces to the 

 Vol.. VIII. Paut IV. 3 I 



