36 Mr Larmor, On Rigidly connected Points, [Jan. 27, 



in the ordinary sense, exist, but there is no occasion to consider 

 them : in place thereof we consider lines which are in the new 

 sense perpendicular to each other, and the theory is an entirely 

 distinct one ; given any two lines, we have perpendicular to 

 each of them (not a single line, but) two lines, or say there are 

 two perpendicular distances : the theory of these distances is con- 

 sidered in some detail. 



(2) A Scheme of the Simidtaneous Motions of a system of 

 Rigidly connected Points, and the Curvatures of their Trajectories. 

 By J. Larmor, M.A., St John's College. 



The following analysis is suggested by the theorems of 

 De la Hire and Savary, whereby the determination of the cur- 

 vatures of the trajectories of the different points of a solid 

 moving in one plane is reduced to geometrical construction. 

 In this theory the construction is based on the circle which at 

 the instant in question is the locus of points for "which the 

 curvature is zero, the well-known circle of inflexions. See 

 Williamson's Differential Calculus, Chapter xix * 



In the generalized theory, when the motion of the solid 

 is not confined to be uniplanar, the first problem is to determine 

 the nature of the locus of inflexions. This is easily effected by 

 kinematical considerations ; for the criterion of a point x, y, z 

 beinsr on the locus is that its acceleration is in the same direction 

 as its velocity, viz. that 



- = ^ = -\. ...(1). 



Now we may specify the motion of the solid by u, v, w the 

 components of the velocity of the origin, and w x , co y , co z the 

 component angular velocities of the body round the axes of 

 coordinates. Then, as usual, 



x = u-yco !l + ZG) v (2), 



x = u — ycb z + zo) y — m z (v — zw x + xco x ) 



+ co y (w-xo) y + ya) x ) (3), 



with two pairs of other similar formulae. 



The equations of the curve of inflexions are now obtained by 

 substitution in (1). 



* I find that questions similar to the ones here discussed are analyzed by the 

 method of vectors from a fixed origin in the Comptes Rendus, 1888, pp. 162 — 5, by 

 Gilbert, who also gives references to other writers on this subject. His investiga- 

 tions relate chiefly to the case when a point of the system is fixed. 



The principal results obtained in this note have been stated in the Cambridge 

 Mathematical Tripos, Part II., June 1, 1889. (Camb. Exam. Papers, 1888-9, 

 p. 569.) 



