ACA-B D first case cp and in the second case cp = Hence if AB, CD be two wires, placed in 2 the focus of the first eye-glass of a telescope, the move able one AB may be made to form every possible an gle with the fixed one CI), and that angle may be rea dily found from the arches AB, CD.
The apparatus by which these arches are measured is represented in Plate CCCLXXVI. Fig. 11, where the graduated circular head may be divided only into 180°, in order to save the trouble of halving the sum, or the difference of the arches AC, BD ; but as it would still be necessary to measure two arches before the an gle could be ascertained, we have adopted another me thod, remarkable for its simplicity, and giving no more trouble than if the wires always intersected each other in the centre of the field.
Let AB. for example, Plate CCCLXXV1. Fig. 9. be the fixed wire, and CD the moveable one, and let it be required to find, at one observation, the angle AOC or p. Let the index of the vernier be at zero, when the point D coincides with B ; and as it is obvious that the extre mity C will be at c when D is at B, the arch c A will be a constant quantity, which we shall call b. Making AC —nz and BD=n, we have, 0= 717 2 n ; but since the ex tremity C will move over the space C c while D de scribes the space DB, these arches must be equal, con sequently b=rn—n hence, adding 2 n to each side of the equation, we obtain b + 2n = m n, or 4_ b+n= in-Fn consequently q=j b+n. Hence the angle AOC 2 is equal to half the arch A c added to the arch DB ; or since A c is invariable, the half of it is a constant quan tity, and the angle required is equal to the sum of this constant quantity and the arch DB.
When the wires do not intersect each other, as in 717 —n Fig. 10, we have 0=— 2 and b=m+n ; hence, sub- tracting 2 n from each side of the equation, we have 77Z-n b —2 n=rn—n, and dividing by 2 4. b—n= 2 , con sequently p=2 b—n. That is, the angle AOB is equal to the difie re.nee between half the arch A c and the arch DB, or to a constant quantity, diminished by the arch DB.
In finding the angle AOB, therefore, we have merely to observe the place of the index when the wires are in their proper position; and as the scale commences at B, or when D and B coincide, and is numbered both ways from B, the degree pointed out on the circular head, when increased or diminished by the constant quantity, will give the angle of the wires which is sought. The
semicircle on each side of a diameter drawn through B, is divided into 18.)°, the 180th degree being at the oppo site end of that diameter.
The method of reading off the angle AOB may be still farther simplified, so as to save the trouble even of recollecting the constant quantity, and of adding and sub tracting it from the arch pointed out by the index of the This effect is produced by making the index of the vernier point to the constant quantity upon the part of the scale below B, Fig. 9, when the points I), B, coincide, or when the wire CD is in the position c B ; for it is obvious that if z is the zero of the scale, and B equal to the constant quantity, the arch D z, whicinis out by the index of the vernier, will be equal to b+n, or the angle MTh. In like manner, in Fig. 10, where the wires do hot cross each other within the field, and where B z is the constant quantity, the arch D z, marked out by the index of the vernier, is obviously equal to 6—n, or the angle A013, which the wires tend to lorm O. By means of tills adjustment, therefore, we are enabled to read off the angle A013 with the same facility as if the wires intersected each other in the very centre of the field, when the arches are accut ate mea sures of the angles at the centre.
It is not necessary that the two wires should be placed in the focus of the first eye-gIsss. Dr. Br ev,ste r constructed an instrument of this kind, in which the fixed wire AB is placed in the locus of the whole piece, or, what is the same thing, in the focus of the ob ject-glass, while the moveable wile CD revolved in the focus of the first eye-glass. In this case the wire AB is more magnified than the other ; but if this should be regarded as an inconvenience, it might easily be removed by using a more delicate fibre.