( 443 ) 



where y.^ and x^ denote the maximum deviations of the pendulums 

 and ;. the length of tlie pendulum synchrone to one of the principal 

 vibrations. 



6. In order to put these equations still more simply, we 



first introduce the lengths of pendulum ^^ = — and ^5= of the 



two suspended pendulums, secondly the maximum deviations in hori- 

 zontal direction of their suspension points : 



(„0 ^ dj^ „,(») ^^^ g (») ^ d^ ,/».) . 

 du\ du 2 



It is then easy to find the following system of equations equivalent 

 to the equations (11), (12) and (13), namely : 

 F{X) = [I'-X) {l,-X) {I,-).) - c,^ I' I, {I -I) - c,^ I' h ih-^-) = ; (14) 



where : 



(mn^ 



(z/'"V 



(15) 



(16) 



We must notice here that c^ and c^ are numerical coefficients, 

 the first of which depends only on the first pendulum and its 

 manner of suspension, the second on the second pendulum. 



Taking note 'of the signification of u' and ^i, and observing that 



for instance §, : zi' '" = 1^ •• u' on account of the supposed small- 

 ness of the vibrations, we can write for the above after some reducing : 



m,(^. * k, ^,§0^ ^'o 



^2_ Ll2 ^ ..2 — iri —(M\ 



Ci — • ^ . ^ — ^ 'I ^ ^ 



m,^^-'^m^^^'-\-U'dm ' rn^^^'' -^ m,^,-" -^ U'^ dm ' 



holding at any moment of the oscillation, where § denotes the hori- 

 zontal, t; the linear deviation out of the position of equilibrium of 

 an arbitrary point of the frame, and where the indices relate to the 

 suspension points 0^ and 0^, whilst the integrations must be extended 

 over the whole frame. 



If we finally remark that the relation between every % and every 

 S is the same as that of the fluxions, we can give the significa- 

 tion of Ci^ and c^^ also in the following words : 



c^"" is equal to the proportion, remaining constant during the motion, 

 between on one side the vis viva of the horizontal motion of the 

 suspension point 0-^ in which the mass of the first pendulum is con- 



31 



Proceedings Royal Acad. Amsterdam. Vol VIII. 



