xxiv TABLE OF CONTENTS. 



CHAPTER. PAGE. 



Ax z +2Bxy+Cy 2 +2Dx+2Ey+F = Q 412 



ax 2 +by 2 +cz 2 = (except x z +y 2 = 2z 2 ) 419 



Further single quadratic equations in n^.3 unknowns 428 



XIV. SQUARES IN ARITHMETICAL OR GEOMETRICAL PROGRESSION 435 



Three squares in arithmetical progression 435 



Four squares in arithmetical progression 440 



Numbers in arithmetical progression all but one being squares 440 



Squares in geometrical progression 441 



XV. Two OR MORE LINEAR FUNCTIONS MADE SQUARES 443 



XVI. Two QUADRATIC FUNCTIONS OP ONE OR Two UNKNOWNS MADE SQUARES 459 



Congruent numbers; x 2 k both squares 459 



Concordant forms: x 2 +my 2 , x 2 +ny 2 both squares. Related problems 472 



x 2 +y and x+y 2 both squares 478 



x 2 +y 2 1 and x 2 y 2 1 both squares 479 



x 2 +2fxy+hy 2 and x 2 +2gxy+ky 2 both squares 480 



Two functions of one unknown made squares 482 



Other pairs of quadratic functions made squares 483 



XVII. SYSTEMS OF Two EQUATIONS OF DEGREE Two 485 



Two quadratic equations in two unknowns 485 



Problems of Heron and Planude; generalizations 485 



System x = 2y 2 -l, z 2 = 2z 2 -! 487 



Number and its square both sums of 2 consecutive squares 488 



Miscellaneous systems of two equations 489 



XVIII. THREE OR MORE QUADRATIC FUNCTIONS OF ONE OR Two UNKNOWNS MADE 



SQUARES 491 



a?x 2 +dx, b 2 x 2 +ex, . . . made squares 491 



Three linear and quadratic functions of x, y made squares 492 



Four quadratic functions of x, y made squares 493 



XIX. SYSTEMS OF THREE OR MORE EQUATIONS OF DEGREE Two IN THREE OR MORE 



UNKNOWNS 497 



x*+y*, x z +z 2 , y 2 +z 2 all squares 497 



Four squares whose sums by threes are squares 502 



Three squares whose differences are squares 505 



Three squares, sum of any two less third a square 507 



Further sets of three or more linear functions of three or more squares made 



squares 509 



Quadratic forms in x and y, x and z, y and z, made squares 511 



xy+a, xz+a, yz+a all squares 513 



Problems related to the last one 515 



Product of any two of 4 or 5 numbers increased by 1 a square 517 



Product of any two of 4 numbers increased by n a square 519 



Other products of numbers in pairs increased by linear functions made 



squares 520 



Further equations whose quadratic terms are sums of products 522 



Squares increased by linear functions made squares 524 



Square of each of three numbers plus product of remaining two a square . . . 526 



Miscellaneous systems of equations of degree two 527 



Rational orthogonal substitutions 530 



XX. QUADRATIC FORM MADE AN WTH POWER 533 



Binary quadratic form made a cube 533 



Binary quadratic form made an nth power 539 



a>iXi z + +a n x, 2 made a cube or higher power 543 



XXI. EQUATIONS OF DEGREE THREE 545 



Impossibility of x 3 +y 3 = 2 3 545 



Two equal sums of two cubes 550 



Three equal sums of two cubes 561 



Solution of 2(z 3 +z 8 ) =y*+t 3 562 



Relations between five or more cubes 563 



Sum of three cubes made a square 566 



Binary cubic form made a cube 566 



