30 SCREW CO-ORDINATES. 



on which is a wrench of given intensity /o". Let the 

 intensities of the components be p/', &c. p 6 /x , and let i\ 

 be any screw. By the principle of 10, a twist about rj 

 must do the same quantity of work acting directly against 

 the wrench on p as the sum of the six quantities of work 

 which would be done by the same twist against each of 

 the six components of the wrench on p. We, therefore, 

 have the equation (using the notation of p. 13) 



P // ^np= P"*^ + &C. + pe'X-.- 



By taking five other screws in place of 77, five more 

 equations are obtained, and from the six equations thus 

 found p/', &c. p 6 " can be determined. This process will 

 be greatly simplified by judicious choice of the six 

 screws of which r/ is the type. Let j/ be reciprocal to w- 2 , 

 &c. w 6 , then *r nua = o &c. ^- na)6 = o, and we have 



f >// ^ P = p/'w,,^. 



From this equation pi" is at once determined, and by five 

 similar equations the intensities of the five remaining 

 components may be likewise found. 



Precisely similar is the investigation which deter- 

 mines the amplitudes of the six twists about the six 

 screws of reference into which any given twist may be 

 decomposed. 



32. The Intensity of the Resultant may be expressed 

 in terms of the intensities of its components on the six 

 screws of reference. 



Let p be any screw of pitch / p , and let p l9 &c. p Q be 

 the pitches of the six screws of reference w b &c. w 6 ; then 

 taking for i\ in ( 26), each of the screws of reference in 

 succession, and remembering that the virtual coefficient 

 of two coincident screws is simply double the pitch, we 

 have the following equations : 



P // ^ pWi = p/^j + p 2 // ^ ft , i(02 -f &C. + p 6 "a-- a 6 

 &C. = &C. 



