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radial peculiar velocity /up (taken positive if the motion is directed from the sun); the mass, m. If now we exclude the sun from the group of stars of which the centre of gravity is rigidly connected with the origin, the equations on page 442 quoted above may be easily reduced to the foliowing form.

2m{ 1 cos8£cos8a)|—2 m cos sin a cos a. tj — 2 m cos S sin S cos a. C = JT 15 sin a cos <S.sin £ cos a.,aa)— 2mcosdcosa

. . 2m( 1—cos8£sin8a)w — JE" 0/cos sin a cos a. £—J£wcos<5sin<5sina.C =


2M(j(— 15 cosacos J.^a + sin^sina.^) —2m cos8sina. /up.

2 nt cos 8$. £ — 2 m cos d sin S cos a. S — 2m cos 8 sin d sin a. r\ — 2 m(j(— cos Sjui) — 2 m sin d.

These equations are rigorous and therefore they determine, free from any hypothesis, the solar motion relative to the adopted origin as a function of the other quantities contained. In order to get his definitive equations *) Bravais introduced certain suppositions in regard to the masses, distances and radial velocities of the stars. By so doing he obtained a form somewhat simpler than the equations (2); but they are no longer a pure expression of the fundamental principle. For this reason we shall provisionally retain the equations in the form (2).

Objections to the method of Bravais.

Evidently the equations in the form (2) contain quantities which are unknown or known only to a small extent, viz:

1. the masses.

2. the distances.

3. the radial peculiar velocities.

Hence there are difficulties in the way of applying the method of Bravais to the determination of the apex. As we have said just now Bravais extricates himself from these by making certain simple suppositions. As in these suppositions lies Bravais' weak point, a just appreciation of the value of his method requires a thorough examination of the difficulties thus introduced. They may be expressed in the form of the two folio wing questions.

1. In how far can we meet these difficulties? On theanswerto thisquestion depends the absolute value of the method.

2. How do the difficulties compare with those arising from the other methods ? The masses and radial peculiar velocities do not enter into the equations furnished by the majority of other methods; the distances either do not appear in them or may be eliminated by a slight modification of the method. Hence, owing to the absence of these quantities, the other methods might seem to be

') Page 446 of the paper of Bravais.