26 REFORT— 1876. 



(=radiu3 of gyration, normal witli respect to a; ; see No. 1), and tlirougli K' draw 

 the perpendicular to K'V meeting MM' in the point K. Tlie straight line passing 

 through K and parallel to / cuts the axis y, conjugate to x (see the construction 

 for it ia No. 2), iu the point L, antipole oi I* ; consequently L is a point on the 

 central nuclexis. 



Oentroids, and their Application to some Mechanical Problems. 

 By Professor A. B. W. Kennedy. 



Elementary Demonstration of a Fundamental Principle of the Theory of 

 Functions. By Paul Mansion, Professor in the University of Ghent. 



M. Thomae (' Abriss einer Theorie der complexen Functionen/ 2'° Aiiflage, Halle, 

 1873, pp. 11-13) first demonstrated rigorously the theorem that "a function ;/=¥x, 

 whose differential coefficient, both iu the positive and in the negative direction, is 

 zero for every value of .r, from Xg to X, is constant iu this iuterval." This important 

 proposition can be demonstrated in an elementary manner by the following method, 

 which seems capable also of other applications. 



I. If the differential coefficient of a function ij = Fx in the positive direction is 



the same as in the negative direction, this differential coefficient is equal, for a 



F(r ) — F(v ) 

 system of values {x, y), to the limit of the ratio ^' '; ^^-^, x„ and x^ converging 



towards the intermediate value x. 

 In fact, by hvpothesis, 



Fx.,-Fx={x.,-x){y'+e;), 



Fa?i— Fa;=(iri-a?)(y'+ei), 

 fi and fj being infinitely small. Consequently 



F.r,-F.r, _ x,-x x^_-x . 



and, fj and tg being multiplied by proper fractions, since x is intermediate to Zy 

 and x^, 



lim 



Fx^-Vx. , 

 — ^ -=u. 



II. Let .To, a*j, . . . . x„_i, X be increasing values of x, to which correspond the 

 values y^, y^, . . . . y„-i, Y of the function 3/= Fa?. 



We have 



Y-yo ^ ('yi-yo)4-(y^-y,) . . . . (Y-y„_i) 

 X-a-o (Xi-x,) + (x^—Xi) .... (X-a;„_i)' 



Y—v 



It results from this equation that -^r—ff- has a value intermediate to the greatest 



and least of the ratios ^'~'^'~\ unless they are all equal. Thus -.— Unless all the 

 Xi — yt-i 



A?/ 



-^'s are equal in the interval (.r„, X), there is at least one of them greater than and 



Y—V 

 one of them less than s? — —. 

 X— a^o 



III. If the differential coefficients of a function y=Fx are the same in the positire 

 direction as in the negative direction, from x^ to X, then either all these diferential 



* In fact if 0L=^ meets I iu L', and if k is the radius of gyration with respect to x iu 



the conjugate direction y, we have, by construction, L'L=L'0+ir^ ; but the distance, in 



the direction y, of the straight line I from its antipole lias exactly this value (see note to 

 my 'Besmni of Eesearches upon the Graphical Eepresentation ' &c.); therefore L is the 

 antipole of I. 



