Analysis of fhe Mecan'iqne Celeste of M, La Place. 37 



to the differential equations of the elliptic motion : the au- 

 thor consequently examines what are the changes to which it 

 is necessary to subject the integrals of the differential equa- 

 tions, in order to have those of the same equations aug- 

 mented by certain terms. The analysis which he uses gives 

 these integrals in a simple and exact manner when the pro- 

 posed equations are linear, and furnishes in general a method 

 of obtaining them by approximation : we may attain the 

 same object by the method of successive substitutions. The 

 author details this process, which has the inconvenience of 

 introducing arcs of a circle, out of the periodical signs in 

 the approximate integral, even when they are not to be 

 found in the exact integral ; and this circumstance takes 

 place when this last should contain, under tlie pcrlodicaJ 

 signs, the very small coefficients, according to the powers 

 of which the approximate integral is ordered : these arcs of 

 a circle being susceptible of an indefinite increase would at 

 length render the approximate integrals defective ; and as it 

 is of importance that these integrals may embrace past and 

 future ages, it is necessary to repass from these arcs of a 

 circle to the functions which produce them bv their deve- 

 lopment into series. In order to attain this object, the au- 

 thor gives a method applicable to any given number of diffe- 

 rential equations; he shows afterwards that the integrals of 

 the differential equations preserve the same form, when these 

 equations are increased by certain terms ; and from this he 

 deduces the means of obtaining, among the arbitrary constant 

 quantities, the conditions relative to tliis last supposition ; 

 he afterwards shows the utility of the variation of the con- 

 stant arbitrary quantities for facilitating the approximate in- 

 tegration of the equations in certain cases*. 



The author applies the preceding methods to the pertur- 

 bations of the celestial motions ; he obtains at first under a 

 titiitc form the periurbations of the motion in longitude, la- 

 titude, and those of the radius vector of the orbit : he after- 

 wards takes up ihe subject of the important problem which 

 has for its object the developmeiit of perturbations in con- 

 vergent series of the sine and cosine of angles increasing pro- 

 portionally to the time. To obtain this, he first gives a very 

 C 3 simple 



