DYNAMICS OF A RIGID BODY. 95 



Xi sin A + pi sin B + v\ sin C = o, 



&c., &c, 

 A 6 sin A + fa sin B + i/ 6 sin C = o. 



From these equations we deduce the following corol- 

 laries : 



The screw of which the co-ordinates are proportional 

 to 0Ai + bfjL ly . . . , a\ + fyc 6 , lies on the cylindroid (A, ju), 

 and makes angles with the screws A, ^u, of which the 

 sines are inversely proportional to a and b. 



The two screws, of which the co-ordinates are pro- 

 portional to a\i bfjiiy . . . , tf A 6 d/j.6, and the two screws 

 A, fj. are respectively parallel to the four rays of a plane 

 harmonic pencil. 



90. Screw Complex of the Fifth. Order and Second De- 

 gree. We have now occasion to make a slight digres- 

 sion from the subject of the present chapter. We have 

 hitherto spoken of the order of a screw complex, and we 

 shall now explain what is to be understood in the use of 

 the word degree. It will be remembered that a screw 

 complex of the $ th order consists of all those screws about 

 which a body having freedom of the 5 th order can twist. 

 We may, however, give an analytical definition of such a 

 complex. It appears from 50 that the six co-ordinates 

 of a screw 9 belonging to a screw complex of the $ th 

 order satisfy that one equation of the first degree 

 which expresses the condition that 6 is reciprocal to the 

 one screw to which the entire complex is reciprocal 

 ( 49). Hence we might with perfect generality define 

 a screw complex of the fifth order and first degree to 

 consist of all those screws whose six co-ordinates satisfy 

 one homogeneous equation of the first degree. 



The reflective reader may be tempted to inquire 

 i into the physical or geometrical meaning of that collec- 



