Prof. Challis on the Problem of Three Bodies. 133 



May 22. — The Lord Wrottesley, President, in tiie Chair. 



The following communications were read: — 

 "On the Application of Photography to the physiouomic and 

 mental phaenomena of Insanity." By HughW. Diamond, M.D. 



The position of the author, as Medical Superintendent of the Surrey- 

 Lunatic Asylum, has enabled him to make the peculiar application 

 of Pliotography, of which he gives an account in the present commu- 

 nication. He points out the advantages to be derived from photo- 

 graphic portraits of the insane, as faithfully representing the features 

 of the disease in its different forms, or its successive phases in the 

 same patient, and as affording unerring records for study and com- 

 parison by the physician and psychologist. In the course of the 

 paper frequent reference is made to the series of photographic por- 

 traits of lunatic patients with which it was accompanied. 



"On the Problem of Three Bodies." By the Rev. J. Challis, 

 M.A., F.R.S., F.R.A.S. 



The object of the author is to give an approximate solution of the 

 Problem of Three Bodies, equally applicable to the motion of the 

 moon and to that of a planet, in which the forms of the develop- 

 ments of the radius-vector, longitude, and latitude in terms of the 

 time, are directly determined by the analysis. The solution to the 

 first power of the disturbing force is effected by means of the follow- 

 ing three equations, in which the letters have the significations 

 usually given to them in the planetary theory : 



•Za.C 



J\dt-\Jdd J dr dt] 

 dt r- r-J( 



dd 

 d^ fis dU_Q 



After substituting in the right-hand side of the first equation, the 

 values of r and d given by a first approximation in which the disturb- 

 ing force is neglected, that side becomes a known function of t. The 

 equation can then be integrated approximately so as to give the de- 

 velopment of r in terms of t to the first power of the disturbing 

 force, and to any pov/er of the eccentricity it may be thought proper 

 to retain. By substituting in that term of the second equation 

 which does not contain the disturbing force the value of r thus ob- 

 tained, the integration of the equation gives the development of 6 in 

 terms of t, and lastly by substitution in the third equation c is simi- 

 larly developed. The author has shown the practicability of this 

 method by obtaining values of r and to terms of the order of the 

 eccentricity multiplied by the disturbing force. The development of 

 the latitude, and a more particular api)lication of the method to the 

 motion of the moon, are reserved for future consideration. The 

 particular advantages of this mode of solution arc, that being free 

 from all assumption as to the forms of the developments, it gives 



