An Elementary Treatise On the Planetary Theory, With a Collection of Problems
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51. The principal part of the coefficient of a term in E, of tJie form P cos { (pn + qn') t + Q} is of the order p + q.
This term arises from the multiplication of such terms as P x C ° S {{Jen - Jen) t+ Q t ), with P 2 1™ {[( P -k)n+(q + Jc) n'] t+Q,}, and P 3 C ° S {[(p + 7c)n+(q- Jc) «'] t + Q 3 ], sm and as in the last Article, the order of the principal part of P will be equal to the lesser of those of P, and P 3 .
48 PLANETARY THEORY.
Now the order of the principal part of P 2 will be the... least value which the arithmetical sum oip~k and q + k can assume, for different values of k.
(i) Suppose k less than p ; then this sum =p —k+q+k =p + q.
(ii) Suppose k not less thanp; then this sum = k —p + q + k, the least value of which (by putting k equal to p) =p + q.
Similarly it may be shewn that p + q will be the order of the principal part of P 3 .
Hence it follows that p + q will be the order of the principal part of P.
In Art. 44 we have assumed that (a 2 + a' 1 — 2aa' cos)~" can be expanded in a series of cosines of and its multiples, we shall now give a proof of this and shew how the coeffi- cients may be calculated.
What to read after An Elementary Treatise On the Planetary Theory, With a Collection of Problems?
You can find similar books in the "Read Also" column, or choose other free books by Charles Hartwell Horne Cheyne to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
51. The principal part of the coefficient of a term in E, of tJie form P cos { (pn + qn') t + Q} is of the order p + q.
This term arises from the multiplication of such terms as P x C ° S {{Jen - Jen) t+ Q t ), with P 2 1™ {[( P -k)n+(q + Jc) n'] t+Q,}, and P 3 C ° S {[(p + 7c)n+(q- Jc) «'] t + Q 3 ], sm and as in the last Article, the order of the principal part of P will be equal to the lesser of those of P, and P 3 .
48 PLANETARY THEORY.
Now the order of the principal part of P 2 will be the... least value which the arithmetical sum oip~k and q + k can assume, for different values of k.
(i) Suppose k less than p ; then this sum =p —k+q+k =p + q.
(ii) Suppose k not less thanp; then this sum = k —p + q + k, the least value of which (by putting k equal to p) =p + q.
Similarly it may be shewn that p + q will be the order of the principal part of P 3 .
Hence it follows that p + q will be the order of the principal part of P.
In Art. 44 we have assumed that (a 2 + a' 1 — 2aa' cos
What to read after An Elementary Treatise On the Planetary Theory, With a Collection of Problems?
You can find similar books in the "Read Also" column, or choose other free books by Charles Hartwell Horne Cheyne to read onlineMoreLess
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