T 



HE 



339 



T H E 



To the student to whom such a direction would be usefu 

 we should say, appeal in all cases to the perceptions de 

 rived from ROTATION. 



To apply the preceding theorems to the statics of a rigid 

 body, we first take the following conventions : Assume an 

 origin and three rectangular axes of co-ordinates, as usual 

 Let the forces which act at each point of the system be 

 decomposed into three, parallel to the axes of x, y 

 and z. Let each force be called positive, when it acts 

 towards the positive part of the axis to which it is parallel 

 if for instance the axis of z be vertical, and if its positive 

 part tend upwards, all forces in the direction of z, wherever 

 they act, are called positive while they act upwards, and 

 negative when downwards. As to couples, let their mo- 

 ments be called positive when, acting in the planes of a; and 

 y, y and z, z and x, they tend to turn the positive part oi 

 the first-named towards the positive part of the second 

 (xy, yz, zx}. Let P, be the first point of the system ; let 

 its co-ordinates be a-,, y,, z, ; let the forces in the three 

 directions acting at that point be X,, Y,, Z,. Let P, be 

 the second point ; ay iy,, z 2 , its co-ordinates ; X 2 , Y 2 , Z 2 , 

 the forces there applied : and so on. All co-ordinates and 

 forces have their proper signs. At the origin apply the 

 following pairs of equilibrating forces, X, and X,, Y, 

 and -Y,, Z, and -Z,; X, and -X,, Y, and - Y 8 , Z 4 and 

 Z 2 , and so on : which of course do not affect the equili- 

 brium, and are over and above those already applied. 

 Again, at the extremity of a",, in the axis of x, apply the 

 equilibrating forces Y,, Y, ; at the extremity of y,, in the 

 if ;/., apply Z,, Z, ; at the extremity of .r,, in the 

 :i\i.- <if z. apply X,, X,, and so on for the other points. 

 Lastly, let the points of application of the original forces 

 X,, Z,, be changed so that each shall act at the projec- 

 tion of the point of application made by its co-ordinate : 

 and the same for the other points. Nothing is done but 

 the application of mutually destroying forces, or the change 

 of the point of application of a force to another point in 

 its direction, and the following figure will show the present 

 arrangement for one point. The original forces, trans- 

 ferred, are marked X, Y, Z ; the original point of applica- 

 tion P, and the other forces, equilibrating two and two, 

 have great and small letters at their extremities. 



n 



We now see that the forces X, Y, Z, are equivalent to 



1. The forces X, Y, Z (marked A, B, C) applied at the 

 origin. 



2. A pair of couples to the axis of z (L, 6) (X, n\ the 

 first positive with the moment YJ-, the second negative 

 with the moment Xy. These two are equivalent to one 

 couple with the moment Yar Xy. 



3. A pair of couples to the axis of x (M, c) (Y, I), 

 the total moment of which is Xy \z. 



-1. A pair of couples to the axis of y (N, a) (Z, m) the 

 total moment of which is Xz-Zz. 



Apply this to every point in the system, and let SX 

 stand for X, + X 2 + , &c., and so on : hence it appears that 

 the whole of the forces are equivalent to forces 2X, sY, 

 Z, applied at the origin in the directions of x, y, and z, 

 together with couples in the planes of ay, yz, ax, of which 

 the moments are 



Xi/X 2(Zy-Y*r), S(X*-Z*). 

 Let A'=2X, L = S(Zy-Yz) 

 B=SY, M=2(X2-/r>, 

 C=SZ, N=2(Yx-Xy) 



_ .... ..^) 



Then it appears that all the forces can be reduced to 

 one force, V, acting at the origin, making angles with the 

 axes whose cosines are A : V, B : V, C : V ; and one couple 

 having a moment W, and whose axis makes with the axes 

 of co-ordinates angles whose cosines are L:W, M:W, 

 N : W. But when there is equilibrium, both the force and 

 the moment of the couple must vanish, for the single force 

 cannot equilibrate a couple. Consequently the conditions 

 of equilibrium are V = 0, W = 0, which give A=0, B=0, 

 0=0, L=0, M=0, N=0, the six well-known conditions 

 of equilibrium. 



The forces will have a single resultant when V falls in 

 the plane of the couple whose moment is W ; that is, when 

 the direction of V is at right angles to the axis of the 

 couple. This takes place when AL + BM + CN = O, a well- 

 known condition. 



For further information we may refer to Poinsot's Sie- 

 mens de Statique ; or, in English, to Pratt's Mathematical 

 Principles of Natural Philosophy ; or Pritchard's Theory 

 of Couples. 



THEORY OF EQUATIONS. Under this term is ex- 

 pressed all that part of algebra which treats of the proper- 

 ties of rational and integral functions of a single variable, 

 such as ax + b, ax^+bx+c, ax* + bx*+cx+e, and so on: 

 a, b, c, &c., being any algebraical quantities, positive or 

 negative, whole or fractional, real or imaginary. Unless 

 however the contrary be specified, it is usual to suppose 

 these co-efficients real, not imaginary. 



The great question of the earlier algebraists was the 

 finding of a value for the variable which should make the 

 expression equal to a given number or fraction : as what 

 must x be so that 3.T*+2.^ may be 11, or x*x?+6x may 

 40, and so on. In modern form it would be asked 

 what value of x will make 3x* + 2x 11 = 0, or x s -x i -}- 

 6x 40 = 0, and so on. To find values of a variable which 

 should make an expression vanish, or become equal to 

 nothing, was then the first desideratum ; and these values 

 are now called roots of the expression. Later algebraists 

 made the finding of these roots subservient to the dis- 

 covery of other properties of the expressions. 



The Hindu algebraists communicated to the Arabs, and 

 hrough them to the Italians, the complete solution of 

 equations of the first and second degrees. The Italians 

 added the solution of equations of the third degree, and 

 of the fourth imperfectly. These last two degrees have 

 >een completed m more recent times, so that it may be 

 now said that the equations of the first four degrees have 

 >een completely conquered : that is to say, having given 

 he equation ax t -{-bx 3 -}-cx t + ex-{-f 0, an algebraical ex- 

 wession can be found, having four values, and four values 

 >nly, and being a function of a, b, c, e, f, which being 

 ubstituted for x on the first side of the equation, shall 

 make that first side vanish. But the student would look 

 n vain through the books of algebra to see this expression : 

 t is both complicated and useless, and it is more desirable 

 o indicate how it is to be found, than to find it. 

 The equation of the fifth degree was attempted in ail 

 uarters, without success : means were found of approxi- 

 lating to the arithmetical value of one or another root in 

 ny one given equation ; but never a definite function of 

 tie co-efficients which would apply in all cases. A proof 

 vas given by Abel, in Crelle's Journal (reprinted in his 

 vorks), that such an expression was impossible, but this 

 iroof was not generally received : it was admitted by Sir 

 W. Hamilton, who illustrated the argument at great length 

 in the ' Transactions ' of the Royal Irish Academy, vol. 

 xviii., part ii. ; but the singular complexity of the reason- 

 ing will probably prevent most persons from attending to 

 the subject. We do not mean in this article to enter intt 

 the history of the theory of equations, but only to place its 

 general state before the reader by an exhibition of the 

 principal theorems, mostly without proof. For works on 

 the subject we may refer as follows: Hutton, Tracts, vol. 

 ii., Tract 33, which contains a full account of the earlier 

 algebraists; Peacock, ' Report on certain Parts of Analysis,' 

 in the Report of the Third Meeting of the British Associa- 

 tion ; or the recent works of Murphy, Young, or Hymers ; 

 all of which are good, and written on such different plans 

 that any one who makes a particular study of the subject 

 will find it advantageous to consult them all. In French 

 the standard works are those of Budan, Lagrange, and 

 Fourier, which however all treat of particular topics ; the 



