by Balance of Errors. 413 



*™e -%-z=^e .% + ™e .coze .f d - y , 



sin#. jr.— s-=©. 

 ad 



This equation is integrablc when multiplied by ^ -, the sola- 



,. • sin 2 £ 



tion gives 2 



Therefore 



sin 



1* e Cm 

 = -tan^.J^.--. 



sin 2 - 



,B S 



This equation is integiable when multiplied by 3-; the solu- 



> a" 



cos 2 - 

 tion gives 3 



1 * 6 



J cos 3 - J sin 2 jr 



/1 

 If - —ty, the solution may be put in the form 



y^-c^'J^'iin^ 5 



^=COt >|r . I dty 



or 



_i_ 1 r„ _© i cot . q . 



^ 3 sim/r.cosi|r J *sin--</r 2 ' "J 'sin 2 ^.* 



C0S 2 iJr J 



y=itanf.J^.^-lcotf.J^.^. 



9. If we substitute for © our value #— sia 0, we obtain 



w=-0-2co4.1ogcos?H-C'.cotan^+C". -Xy. 

 * 2 ° 2 2 sm 



(in which we may vary the arbitrary terms, remarking that 



-■— a =o tan o + o cot o) » 

 sm 6^ 2 2 2 2/ 



aud if we determine the arbitrary constants so that, when 0=0, 



