Elements of natural philosophy (Volume 2-3) . ear- ^ we attribute a slight change to one of the disturbingcauses, the corresponding change in the displacementmust be proportional thereto; and whether the changein all the partial causes, or in the functions whichmeasure them, be simultaneous or successive, the finalresult will be the same; for, the change in the entirefunction in the first case must be equal to the algebraicsum of the partial changes in the second. To those fa-miliar with the calculus, it wTill be sufficient to say, thatthe first power of the total differential of the sum of an
Elements of natural philosophy (Volume 2-3) . ear- ^ we attribute a slight change to one of the disturbingcauses, the corresponding change in the displacementmust be proportional thereto; and whether the changein all the partial causes, or in the functions whichmeasure them, be simultaneous or successive, the finalresult will be the same; for, the change in the entirefunction in the first case must be equal to the algebraicsum of the partial changes in the second. To those fa-miliar with the calculus, it wTill be sufficient to say, thatthe first power of the total differential of the sum of anumber of functions, is always equal to the first powerof the sum of the partial differentials. We conclude, therefore, that the function which givesthe displacement may be broken up, so to speak, intoseveral partial functions equal in number to that of thedisturbing causes ; that these j>artial functions will besimilar to each other and to the entire function; andthat this latter wTill be equal to the algebraic siun ofthe former. Fig. § 57. To illustrate :let the straight lineA B, be the locus of a illustration; series of molecules intheir positions of rest;the fine waved linea 5, that of the samemolecules at a particu-lar instant of time, when disturbed and thrown into awave by the action of some single cause ; and the waved Partial ^aves; line a b, that of the same molecules at the same in-stant had they been thrown into a different wave underthe operation of some other insulated action. If thesedisturbing causes had acted simultaneously, the locus of construction of the disturbed molecules would be represented by the resultant wave. r J heavy waved line X Z, constructed in this wise : Atthe various points of the line A B, erect perpendiculars ELEMENTS OF ACOUSTICS. 63 and produce them indefinitely; lay off from A B, on Resultant curve;these perpendiculars, distances equal to the sum or dif-ference of the corresponding ordinates of the componentcurves, according as t
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