## AN INTRODUCTION TO DETERMINANTS WITH NUMEROUS EXAMPLES |

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&ift a2 ft a2 h a2ax algebraic algebraic function av a2 ax a2 ax bx ax h axa2 azAz complementary minors complete square conjugate conjugate elements constant terms cubic equation cx dx cxc2 degree determinant vanishes dexter diagonal elements example factor fourth order ft 7i ft ft function given determinant h ax h C2 h C4 h d2 h dx H h ds h h h Hence homogeneous homogeneous functions interchange leading diagonal leading term minant multiplied nth order pairs of rows reducing respectively row or column Sabc second column second minors second order second row similar manner Similarly skew symmetrical determinant suffixes system of equations theorem Theorem.—If third order tion treatise true for determinants variables x y z

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Page 34 - II, 1881, pp. 491-535. 5. The determination of the secular effects of tidal friction by a graphical method. < Proceedings of the Royal Society of London, vol. 29 (1879), pp. 168-181. Darwin's method of treatment is to express the tide-generating potential as a sum of terms, each of which is the product of a second-order solid harmonic and a simple time harmonic, and then to derive the corresponding surface harmonics which define the tidal deformations when the system has assumed a condition of steady...

Page 57 - ... the product of two numbers, each of which is the sum of four squares, can be represented as the sum of four squares.

Page 3 - A polynomial is said to be homogeneous, when all its terms are of the same degree.

Page 51 - In a similar manner we can express the product of two determinants of the third order as a determinant of any order from the third to the sixth inclusive, and similarly for determinants of the nth order.

Page 21 - Theorem. — -A determinant remains unaltered, when we add to the elements of any row or column the corresponding elements of any of the other rows or columns multiplied respectively by constant factors. Thus Д = Ъп сп . . . kri + + pbn+ +¿kn Ъпсп . . . kn but the second of these determinants vanishes by Art.

Page 56 - The notation (where the number of columns is greater than the number of rows) is used to express the three determinants which can be obtained by suppressing in turn each one of the columns ; viz. the three determinants of which we have been speaking, (aa68), («a61), («A)

Page 20 - ... one column, can be expressed as a single determinant. This is merely the converse of the preceding theorem, and can be proved at once by reversing step by step the proof there given. We leave this to the student.

Page 46 - Q of the second order can be expressed as a determinant of the second order in four ways in general different.