<6o THE PRINCIPAL SCREWS OF INERTIA. 



Whenever a given external wrench is replaced by an 

 equivalent wrench upon a screw of the complex which 

 .defines the freedom of the body, the latter may be termed, 

 for convenience, the reduced wrench. 



It will be observed, that although the reduced wrench 

 can always be determined from the given wrench, that 

 the converse problem is indeterminate (n < 6). 



We may state this result in a somewhat different 

 manner. A given wrench can always be resolved into 

 two wrenches one on a screw of any given complex, 

 and the other on a screw of the reciprocal screw com- 

 plex. The former of these is what we denote by the 

 reduced wrench. 



67. Co-ordinates of Impulsive and Instantaneous 

 Screws. Taking as screws of reference the n principal 

 screws of inertia ( 57), we require to ascertain the rela- 

 tion between the co-ordinates of a reduced impulsive 

 wrench and the co-ordinates of the corresponding instan- 

 taneous screw. If the co-ordinates of the reduced wrench 

 are n/', . . ., rj n x/ , and those of the corresponding twisting 

 motion are a/,. . , a,/, then, remembering the property 

 of a principal screw of inertia ( 57), and denoting by 

 MI, . . ., u n , the values of the magnitude u ( 59) for the 

 principal screws of inertia, we have, from 60, 



whence we deduce the following theorem, which, in the 

 particular case of n = 6, reduces to that of 53. 



If a quiescent rigid body, which has freedom of the 

 n th order, commence to twist about a screw a, of which 

 the co-ordinates, with respect to the principal screws of 

 inertia, are ai, . . . a n and if/i, ...,/ be the pitches, 

 and u ly . . ., u n the constants defined, in 59, of the 



