A Treatise on the Theory of Screws

486 THE THEORY OF SCREWS. In the case of n = 4 the function T will consist of 10 terms such as An0I2+... + 2A1ff2 + .... If any arbitrary values be assigned to ^JS| &c., it will still be possible to determine a rigid body such that this function shall represent w/ (to a constant factor), because we have 9 co-ordinates disposable in the rigid body. Hence for n = 4 and a fortiori for any value of n less than four the function representing T will be a function in which the coefficients are perfectly unrestricted. Hence n = or < 4 the determinantal equation is in our theory of the most general type. The general theory while affirming that all the roots are real does not prohibit conditions arising under which roots are repeated. Hence Routh’s important theorem becomes of significance in cases n= 2, n = 3, n = 4 for in these equations the roots may be repeated. But in the case of n = 5 the function T consists of 15 terms. If arbitrary values could be assigned to the coefficients then of course the general theory would apply alld cases of repeated roots might arise. But in our investigation the 15 coefficients are functions of the co-ordinates which express the most general place of a rigid body, and these co-ordinates are not more than nine. If these nine co- ordinates were eliminated we should have five conditions which must be satisfied by the coefficients of a general function before it could represent the T of our theory even to within a factor. The necessity that the coefficient of T shall satisfy these equations imports certain restrictions into the general theory of the deter- minantal equation based on T. One of these restrictions is that T shall have no repeated roots. The same conclusion applies a fortiori to the case of n = 6. The subject may also be considered as follows. Let us first take the general theorem that when reference is made to n principal screws of inertia of an «-system the co-ordinates of the impulsive wrench corresponding to the instantaneous screw For a principal screw of inertia the ratios must be severally equal or uf ctj = u/ a.2 = = uf q,( Pl “1 Pi <h Pn af These equations can generally be only satisfied if n - 1 of the quantities “i ... an, be zero, i.e. there are in general no more than the n principal screws of inertia. If however u2 _ uf Pi P2’ / then though we must have a3 = 0...am = 0, dj and a2 remain arbitrary.